Analysis of a Discrete First-Order Model Reference Adaptive Controller Discretized by the Zero-Order-Hold Discrete Equivalent

4th World Conference on Structural Control and Monitoring  

4WCSCM-066

ANALYSIS OF A DISCRETE FIRST-ORDER MODEL REFERENCE ADAPTIVE CONTROLLER DISCRETIZED BY THE ZERO-ORDER-HOLD DISCRETE EQUIVALENT O.S. Bursi, N. Tondini and L. Vulcan University of Trento, Trento, 38050, ITALY [email protected]   , [email protected]   , [email protected] 

D.P. Stoten University of Bristol, Bristol, BS8 1TR, UK [email protected]  Abstract

Structural active control has ever greater interest, as it allows increasing the performance and the life of a structure in the face of environmental actions. In detail, the adaptive control involves techniques that permit to control systems whose characteristics are uncertain or can vary with time. Nowadays, control systems are digitally implemented, then the control algorithm must be written in a discrete form. The research made has the aim to analyse a model reference adaptive controller (MRAC) discretized by the zero-order-hold (ZOH) discrete equivalent. In an MRAC the characteristics of the system are given in terms of a linear reference model and the goal of the controller is to match the response of the plant with that of the reference model by means of an adaptive mechanism conceived to compensate for system nonlinearities. Among the MRAC controllers, the Minimal Control Synthesis (MCS) algorithm is analysed and applied for instance to a linear first-order SDoF system. The single step/stage ZOH method is used, as it is suitable to be implemented for real-time applications. Both stability and accuracy analyses are carried out in order to identify the performance of the MCS algorithm. The analysis is conducted by system linearization at steady-state conditions through a physical insight  approach.  approach. The unavoidable presence of a delay in the computation of the control law leads to consider its effect by carrying out the comparison of two cases: (i) no delay; (ii) delay equal to one time step. Finally, numerical findings are validated by means of simulations.

Introduction

The use of adaptive control techniques has become a topic of increasing interest in recent years as adaptive control can be used to control plants whose parameters are unknown or uncertain (Åstrom et al., 1995). In the context of control theory some recent books and papers relevant to direct methods can be found (Sastry, 1999; Wagg, 2003), while for structural control several applications have been made (Housner et al., 1997). When a direct reference model controller is used, it can be also applied to systems where the details of the plant cannot be fully known a priori   or are time varying. Using this type of algorithms assuming zero initial conditions for the controller gains, i.e. without the need of the plant  parameter knowledge, has become known as the Minimal Control Synthesis (MCS) approach (Stoten and Benchoubane, 1990). Basing adaptive control schemes on a reference model enables the controlled system to behave like the model itself. This type of approach has been applied to a wide range of plants including nonlinear systems (Wagg, 2003). As the approach is primarily based on linear control theory,  being the reference model usually a linear one, the effect of nonlinearities and/or disturbances in nonlinear systems is compensated for by the adaptive nature of the controller. Nowadays, controllers are implemented on digital computers, and therefore, control systems must be directly written in digital form and/or transformed from continuous time to discrete time; moreover, if possible, they must run in realtime and compensate for the influence of time delay (Chu et al., 2002). Particular attention will be given to the time delay issue because it sensitively affects the control system behaviour and stability. In general, the total time delay in a control system can be divided in two contributions. The first part is referred to as fixed time delay due to the on-line data acquisition, filtering, manipulation of digital data inside the digital control processor, calculation of required control force and signal transmission from the computer to the actuator. The second part usually depends on the particular dynamics of the actuators interacting with the controlled system, which is referred to as floating delay time that can be adjusted by explicitly combining the actuator dynamics with that of the structure (Chu et al., 2002). Besides, time delays are

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often the source of instability of the system (Xu and Lam, 2005), so this paper will investigate the  behaviour of the system by considering one time step delay in the discrete control law. The selection of the best sample rate for a digital control system still represents a problem. Thus generally, the slowest sample rate that meets all performance specifications is sought. The paper is organised as follows. Initially, the MCS algorithm formulation and the adaptive mechanism for nonlinearity compensation will  be presented. Then, the MCS algorithm with one time step delay in the control law, discretized by implementing the ZOH integrator, will be investigated by means of stability and accuracy analyses. Afterwards, simulations capable to show the effect of the delay on the performance of the control system will be presented. Finally, the main conclusions along with comments on future perspectives will be drawn. The Minimal Control Synthesis algorithm

This section introduces the features of the adaptive MCS controller exploited in this paper; this controller is characterized by an adaptive portion, conceived to compensate for system nonlinearities. A generic nonlinear plant can be described by the following equations: x (t ) = Αx(t ) + Bu(t ) + f NL (x(t ), t )

 

(1)

y (t ) = Cx(t )

where A is the n×n plant matrix, B is the n×m input matrix, x is the n-dimensional state vector, u is the mdimensional control input vector and f  NL represents the nonlinear and time varying terms. y and C are the output vector and plant output matrix, respectively, each having appropriate dimensions. A linear system with the same number of states, inputs and outputs is selected as reference model. The goal of the controller is to match the characteristics of the actual nonlinear system with those of the reference model. The reference model has the following form: x M (t ) = Α M x M (t ) + B M r (t ) y M (t ) = Cx M  (t )

 

(2)

where AM and BM are n×n and n×m constant matrices, respectively, xM is the reference model state vector, r is the m-dimensional reference input vector, and yM is the reference model output. The error vector, xE, can be defined as the difference between reference model, xM, and plant state, x. The desired result for the controller is to drive the steady-state error such that lim t→+∞xE=0, when this limit exists. If the plant is time varying or nonlinear, then the f  NL  vector influences the controller. For this reason, the Model Reference Adaptive System (MRAS) theory introduces a time varying controller: u ( t ) = - ( K - δ K ( t ) ) x ( t ) + ( K R + δ K R ( t ) ) r ( t )  

(3)

 

where δK   and δK R   are time varying gain adjustments. Substituting the control law (3) in (1) and after some manipulations, we get x E ( t ) = A M x E ( t ) + ( A M  - A + BK ) x ( t ) +

+ ( B M

(

- BK R ) r ( t ) - B (δ K R ( t ) r ( t ) + δ K ( t ) x ( t ) ) + f NL ( x ( t ) , t )

)

 

 

(4)

K  and K R  are selected so that the second and third terms in (4) are cancelled out. The error equation now

reduces to: x E ( t ) = A M x E ( t ) - B (δ K R ( t ) r ( t ) + δ K ( t ) x ( t ) ) - f NL ( x ( t ) , t )  

Bursi, Stoten, Tondini and Vulcan

(5)

2

The values of δK   and δK R  must be selected to cancel out the nonlinear terms, but they cannot explicitly solved because x  is unknown. A stability proof for these controllers has been developed by using the hyperstability theory (Landau, 1979), which deals with the stability of systems that can be broken into a linear feed-forward loop, which meets the strictly positive real condition, and a nonlinear feedback loop that satisfies the Popov criterion for hyperstability. This problem is solved by the proper selection of the output error matrix CE, used to define the output error yE(t) = CE xE(t), according to the Lyapunov  problem. A solution that satisfies the hyperstability condition yields expressions for δK  and δK R (Landau,   1979):

δ K R

t 

∫ α y (τ ) x (τ ) d τ + β y   (t ) x (t ) ( t ) = ∫ α y (τ ) r (τ ) d τ + β y   ( t ) r ( t )

δ K (t ) =

T

T 

 E

0

E 

t 

T

E

0

(6)

T 

E 

where α and β are two positive constants, that must be selected before the solution of the above equations. In order to ensure the stability of the closed loop system, AM must have eigenvalues in the left hand side of the complex plane and should also have a Luenberger-type controllable canonical structure to ensure that Erzberger’s conditions are satisfied (Landau, 1979). The adaptive controller presented above requires the knowledge of the plant characteristics, A and B, which allows evaluating the constant values of K  and K R.  This drawback can be eliminated by using the MCS algorithm which derives from the MRAS formulation, but it assumes zero initial gains K   and K R   and, consequently, unknown plant parameters (Stoten and Benchoubane, 1990). Figure 1 shows the block diagram of the MCS controller with its essential components: the adaptive block; the input signal generation, u; the plant; together with a parallel reference model; and a common reference signal, r. The only elements that are needed to know are: the reference model parameters AM and BM; the structure of the plant with the degree of freedom and order and the sign of the coefficients of B, that usually is assumed positive. Therefore, the control law becomes u ( t ) = K ( t ) x ( t ) + K R ( t ) r ( t )   

(7)

For the value of the output matrix CE, Stoten and Neild (2003) proposed a pragmatic solution for firstorder (CE=[4/ts]) and second-order ( CE=[4/ts 1]) one Degree-of-Freedom systems (DoFs), where t s is the time settling of the reference model. This solution entails an exact pole-zero cancellation. xm

C

xm=A mxm+B mr 

+ r 

+ +

xe

u=Kx+K R r  x=Ax+Bu

x

Ce C

ym ye y

Figure 1. Block diagram of the classical Minimal Control Synthesis algorithm.

Discrete formulation by means of the ZOH method

Since usually the controller is implemented in a digital fashion, the continuous controller developed in the  previous section must be discretized. This transformation is performed by means of the ZOH sampling  process which exhibits several features among which, exact solution at sampling points for linear time

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invariant (LTI) systems and real-time compatibility. The discrete-time form of the reference model is defined by the mapping x M [ k + 1] = ΦM x M [ k ] + Γ M r [ k ]  

(8)

resulting in one-step one-stage method, where ∆t 

Φ

=e

∆t A M 

  and

Γ M 

= ∫ eτ A

 M 

d τ B M   

(9)

0

The sampling period or time interval is ∆t= t k+1 - tk  where the discrete time variable is the integer k. The discrete control law equation for the MCS controller reads: u [ k ] = K [ k ] x [ k ] + K R [ k ] r [ k ]  

(10)

with the adaptive gains, K [ k ] = K [ k − 1] + β y E [ k ] x

[ k ] − σ y E [ k − 1] xT  [ k − 1]   T  T  K R [ k ] = K R [ k − 1] + β y E [ k ] r [ k ] − σ y E  [ k −1] r [ k − 1] T

   

(11)

where σ=β−α∆t. For the assumptions of the MCS algorithm, the initial conditions read K [−1]=0  and K R[  −1]=0. In order to complete the characterization of the MCS controller, the eigenvalues of the reference model, α and β from the adaptation equations, just as CE must be selected. The reference model must be stable, i.e. the eigenvalue moduli must be less than one, and should be selected so as not to exceed the system capability. The values of α and β are arbitrary positive numbers and are selected by trial and error. However, an increase of β means a reduction of the settling time of the adaptation, while a reduction of the β/α ratio improves the damping. The ratio β/α can be set to 0.1 as compromise between the speed of adaptation mechanism and stability limit (Vulcan, 2006). The final values of α and β are not critical, but they cannot be increased indefinitely because that may magnify noise within the loop. The selection of the error vector weighting matrix CE is also relatively arbitrary and the selected values have  been already introduced in the previous section for the continuous time system. Due to the presence of the adaptive process, the MCS algorithm defines a nonlinear dynamical system; therefore, for simplicity, we will consider first order linear systems only both for the plant and the reference model, and we will  perform the analysis on SDoF model equations. As a result:  x [ k + 1] = A ' x [ k ] + B ' u [ k ]

  ' ' k A x k B r k    + 1 = + ] M M [ ] M  [ ]  M [

(12)

' ' where:  A' = e −∆ t / T  ,  B ' = 1 − e −∆ t / T  ,  A M  = 1 − e−∆t / T   owing to the ZOH sampling; T and = e−∆t / T  , and  B M  TM are the plant and the reference model time constants, respectively; the low frequency gain for the plant and the model are assumed equal to 1; and T M is related to the approximate settling time t s by TM=ts/4. Furthermore, the control signal and the reference input, that is assumed as a step function, read,  M 

 M 

u [ k ] = K [ k ] x [ k ] + K R [ k ] r [ k]    r [ k ] = 0 for k < 0, r [ k ] = 1 for k

(13)

≥ 0   

(14)

respectively. In (13), we have implicitly assumed that the processing time δ is negligible with respect to the time step t and, therefore, the control signal u, generated by the MCS algorithm, is obtained instantaneously and held constant over t. Furthermore, (11) are written in scalar form and CE= 4/ts is selected according to the proposed solution. As just stated, the control variable u[k] and the state variable

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x[k] are assumed to be computed at the same time. Evidently, this is an approximation as the time delay during the evaluation of the control signal is unavoidable. In this sense the behaviour of the system will  be analysed by considering one time step delay in the discrete control law and, consequently, the comparison between the case with and without delay will be presented. In these conditions, the control law of the MCS controller reads, u[k ] = K [ k − 1]x[k − 1] + K R [ k −1]r[ k −1 ]

(15)

We emphasize that (15) entails another approximation, since in reality the delay may be a fraction of t; however it is always possible to manage the delay to be equal to the time step, such that (15) results to be correct. Therefore, the analyses provided hereafter allow the effect of a delay in the control law computation on the system to be understood. Stability and accuracy analysis

Following Vulcan (2006) the discrete-time equations that rule the system with delay can be arranged as: ⎡  A' ⎧ x E [ k +1] ⎫ ⎧ xE [ k ] ⎫ ⎧ B 'm r[ k ]⎫ 0 ⎢  M ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −  x k x k  1 ⎢ 0 [ ] [ ] 1 0 ⎪  E ⎪ ⎪ E  ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎨  x[ k +1] ⎬ = C MD ⎨ x[ k ] ⎬ + ⎨ 0 ⎬ , C MD = ⎢ 0 0 ⎪  K [ k ] ⎪ ⎪ K [ k −1] ⎪ ⎪ 0 ⎪ ⎢ β x[ k ]C −σ x[ k −1]C   E E  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪⎩  K R [ k ] ⎪⎭ ⎪⎩ K R [ k −1]⎪⎭ ⎩⎪ 0 ⎭⎪ ⎢⎣ β r[ k ]C E −σ r[ k −1]CE 

' − A' AM  0  A ' 0   0

− B ' x[ k −1] − B ' r[ k −1]⎤⎥ ⎥ 0 0 ⎥ B ' x[ k −1] B ' r[ k −1] ⎥   ⎥ 1 0 ⎥ 0 1 ⎥⎦

(16)

In order to reduce the complexity of the analysis, it will be locally developed by linearizing the dynamical system by a  physical insight   approach around an operating point. The linearization by  physical insight  is simply carried out by substituting the values of the operating point in the amplification matrix, which in turn is arranged in a way to reduce the complexity of the analysis. One can note that the system linearization could be obtained through a more rigorous linearization procedure based on a Taylor series expansion about the operating point (Dutton et al., 1997), but this approach is more burdensome than the  physical insight because it requires preliminary simulations in order to estimate K[k] at steady state. Considering a unit step input, the operating point at steady-state conditions reads: x E[k-1]=xE[k]=0, x[k1]=x[k]=1, r[k-1]=r[k]=1. In this way, substituting the operating point in the amplification matrix (16), we obtain the amplification matrix at steady state.

C MD,ss

⎡  A M' 0 ⎢ 0 ⎢ 1 =⎢ 0 0 ⎢ ⎢ β C E −σ C E  ⎢⎣ β C E −σ C E 

' AM  − A'

0  A '

0 0

− B ' − B '⎤ ⎥ 0 0 ⎥   B' B' ⎥ ⎥ 1 0 ⎥ 0 1 ⎥⎦

(17)

Firstly, CMD,ss is assembled in order to avoid having terms depending on x E, as they would become null once substituted the operating point in CMD,ss. Secondly, the operating point contains the steady-state values of K and K R  that are unknown, so every term exhibiting the values of K and K R  is moved to the augmented vector at step k. These choices allow an amplification matrix at steady-state to be obtained easy to be analysed by introducing the following non-dimensional variables: t/TM, β/(4TM), T/TM and β/(α4TM). It can be proved that the stability of the system is assured when the modulus of the eigenvalues is less than one (Bursi et al., 2006). Considering that it is possible to determine the Sampling interval Gain (SG) space, which represents the asymptotic stability regions by varying t/TM  and β/(4TM), for β/(α4TM)=0.1 and r[k]=1, as it is depicted in Figure 2. These spaces may also be interpreted as operational tools that allow the optimal combination of t-β/α values to be selected; conversely they are usually set  by trial and errors, even if the T/T M ratio is unknown. The advantage of these spaces is that they permit

Bursi, Stoten, Tondini and Vulcan

5

the operators to determine a priori   information on the behaviour of the sampled-data control system. From Figure 2 it is evident that, if the control law u[k] is computed with one time step delay, the stability limit reduces. Such a result was expected as the MCS controller is not able to promptly modify the system  parameters due to the delay in the evaluation of the control law. These findings underline the need of reducing the time of the control law evaluation or to assume small ratios t/TM, either by using small t or large TM, to increase the stability of the system for delay effects. The results found on system stability are confirmed by the plot of the SG space shown in Figure 2. Moreover, it highlights a greater reduction of the stability regions when the ratio T/TM increases. An increase of the ratio T/TM means that the plant  becomes ever slower, hence, a loss of time in the control law computation, due to the delay, is more  burdensome in terms of stability. It may be also observed that the complex limit of the system without delay, which is independent from the plant characteristics, disappears; hence, the stability boundaries depend on the plant. 100

100

 / (4T m)

 /(4Tm) 10

10 1

0.1 1 0.01

0.1

0.001 0

0.1

0.2

T/Tm= 1 d elay T/Tm= 10

0.3

0.4

0.5

0.6

0.7

T/Tm= 10 de lay c omplex c on ju gate

0.8

0.9

1

0.001

t/Tm

T/Tm=1

Figure 2. The SG space for the MCS with one time step delay, r[k]=1 and β /(α4TM)=0.1.

0.01 T /Tm =1 de lay T /Tm =10

T/Tm=10 delay comp le x c onjug ate

0.1 T/T m=1

1

 /( 4Tm)

Figure 3. The GS domain for the MCS with one time step delay, r[k]=1 and ∆t/TM=0.1

An attentive reader may observe from Figure 2 that the behaviour of the system with delay starts to be quite independent from the plant characteristics when t/TM approaches 0.4 and from this value onwards.  Notice that 0.4 corresponds to the vertical asymptote for the system without delay. Similar conclusions can be drawn from the Gain Space (GS) domain for variable β/(α4TM) and t/TM=0.1 depicted in Figure 3, where the system behaves independently from the ratio T/T M as long as β/(α4TM) ≤ 0.025; this value corresponds to the vertical asymptote of the complex limit for the system without delay. It is worthwhile  presenting also the algorithmic damping ratio ξ   and the algorithmic damped numerical frequency Ω , as defined by Bursi et al.  (2006), in order to show how the MCS algorithm influences the system  performance. It may be observed from Figures 4 and 5 that both ξ   and Ω  introduced by the controller decrease for the same parameter combination of Figure 2. − ξ

− Ω

2. 5

β  

4 Tm

β 

= 1   del ay

4 Tm

β 

4 Tm

π

= 1

= 1   del ay

β  =  10   del ay

4 Tm

β   =  10   del ay

2

β  =  10

4 Tm

4 Tm

β  

4 Tm

= 1

β   =  10

4 Tm

3 π  4

1. 5 π 

2

1 π 

4

0. 5 ∆t  Tm 

0. 2

0. 4

0. 6

0. 8

1

Figure 4. ξ   as function of t/TM for the MCS algorithm with one time step delay, T/T M=10 and r[k]=1.

Bursi, Stoten, Tondini and Vulcan

0. 2

0. 4

0. 6

0. 8

1

∆t   Tm

Figure 5. Ω  as function of t/TM for the MCS algorithm with one time step delay, T/T M=10 and r[k]=1.

6

Performance of the MCS algorithm with one time step delay

In order to study the performance of the MCS algorithm with one time step delay, the mean square tracking error XE[k] and the tracking error bound X E,sup[k] are used. They are suggested and defined in Bursi et al. (2006). From Figure 6, one can note that X E increases faster for the MCS with delay when t/TM becomes large; this is particularly evident when t/Tm approaches 0.4. This behaviour is due to the fact that the system with delay at that value is getting unstable, see Figure 2, and, therefore, it is much closer to the stability limit than the standard MCS. This trend can be also clearly observed in Figure 7 for T/Tm=10, this figure reports the tracking error bound X E,sup as function of t/Tm. xE

0.01

0.1

1

1.E+00

xE,sup

0.01

0.1

1

1.E+01

1.E-01

1.E+00

1.E-02 1.E-01 1.E-03 1.E-02 1.E-04 1.E-03 1.E-05 1.E-04

1.E-06

1.E-05

1.E-07 1.E-08

1.E-06

t/Tm

Figure 6. Convergence of XE at t=ts for the MCS with one time step delay, T/TM=10 and r=1: the full line correspond to β /(4TM)=0.1; the dashed to β /(4TM)=1; the dotted to β /(4TM)=10.

t/Tm

Figure 7. Convergence of X E,sup at t=2ts for the MCS with one time step delay, r=1 and T/T M=10: the full line corresponds to β /(4TM)=0.1; the dashed to β /(4TM)=1; the dotted to β /(4TM)=10.

Representative numerical simulations

The simulations presented hereafter intend to validate some of the analytical findings. It is demonstrated that the MCS with one time step delay determines a reduction of the stability limits as well as the algorithmic damping introduced by the controller (Vulcan, 2006). For instance, the evolution of the MCS controlled system with and without delay is depicted in Figures 8 and 9, respectively, assuming β/(4TM)=10 and t/TM=0.1. They highlight that the system controlled by the MCS with delay starts to  become unstable, while the system controlled by the standard MCS keeps staying stable. This could be verified from the SG space of Figure 2. Moreover, Figure 8 shows how the system is less damped out and the oscillations become larger, due to a reduced value of the algorithmic damping depicted in Figure 4. Conversely, the system without delay depicted in Figure 9 does not exhibit any particular change in this respect. Conclusions and perspectives

This paper has dealt with the convergence and the steady-state analysis of a discrete first-order MCS controller sampled with the one-step one-stage ZOH discrete equivalent. In detail, it was assumed that the  plant to be controlled was a first-order system and the communication between the plant and the controller was zero-order sampled. In the proposed analysis, the control signal was generated considering the effect of a time delay, at this stage equal to one time step t, as the delay is always present in physical devices. The analysis was performed for the case where the demand corresponds to a unit step input. Due to the system nonlinearity, the system was linearized by means of a novel  physical insight  approach. This analysis allows two gain space domains to be determined in order to define the region of local stability. Moreover, the accuracy analysis has provided insight into the range of adaptive control weights that result

Bursi, Stoten, Tondini and Vulcan

7

in optimal performance of the MCS controller. This analysis also highlighted a possible approach to a  priori selection of the time step and adaptive weighting values. 14

14

12

12

10

10

8

8

6

6

4

4

2

2 0

0 0.0

0.5

1.0

1.5

2.0

0.0

2.5

-2

-2

-4

-4

0.5

1.0

1.5

2.0

2.5

-6

-6

x

xe

K

Kr

 

Time

Figure 8. Evolution of the MCS controlled system with delay for r[k]=1, T/TM=10, β /(4TM)=10, β /(α4TM)=0.1, and ∆t/TM=0.1.

x

xe

K

Kr

 

Time

Figure 9. Evolution of the MCS controlled system for r[k]=1, T/TM=10, β /(4TM)=10, β /(α4TM)=0.1, and ∆t/TM=0.1.

In terms of future perspectives, the analysis proposed in this paper and performed for a first-order system, could be developed for second-order systems, which are typical present in structural systems, in order to complete the convergence and the steady-state analysis of the discrete MCS controller. The effect of the delay reduces the stability limit and the effectiveness of the numerical damping; this effect represents an important characteristic of discrete control system that can be further deepened in control problems considering the effect of variable delay. Acknowledgements

The authors thank the Department of Mechanical and Structural Engineering for the SMART STRUCTURES project, that sponsored the research project by means of grants. However, opinions expressed in this paper are those of the writers, and do not necessarily reflect those of the sponsor. References

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