Robust Control and Time-Domain Specifications for ... - CiteSeerX

Sun Yi1 Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125 e-mail: [email protected]

Patrick W. Nelson Center for Computational Medicine and Biology, University of Michigan, 100 Washtenaw Avenue, Ann Arbor, MI 48109-2218 e-mail: [email protected]

A. Galip Ulsoy Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125 e-mail: [email protected]

1

Robust Control and Time-Domain Specifications for Systems of Delay Differential Equations via Eigenvalue Assignment An approach to eigenvalue assignment for systems of linear time-invariant (LTI) delay differential equations (DDEs), based upon the solution in terms of the matrix Lambert W function, is applied to the problem of robust control design for perturbed LTI systems of DDEs, and to the problem of time-domain response specifications. Robust stability of the closed-loop system can be achieved through eigenvalue assignment combined with the real stability radius concept. For a LTI system of DDEs with a single delay, which has an infinite number of eigenvalues, the recently developed Lambert W function-based approach is used to assign a dominant subset of them, which has not been previously feasible. Also, an approach to time-domain specifications for the transient response of systems of DDEs is developed in a way similar to systems of ordinary differential equations using the Lambert W function-based approach. 关DOI: 10.1115/1.4001339兴

Introduction

A primary goal for control engineers is to maintain the stability of a system, an essential requirement, while achieving good performance to meet response specifications 关1兴. For a system of delay differential equations 共DDEs兲, even though more complex than systems of ordinary differential equations 共ODEs兲 due to its transcendental characteristic equation, various methods to achieve that goal have been introduced in the literature in recent decades. For a detailed survey, refer to Refs. 关2,3兴 and the references therein. However, systems frequently have uncertainties in model parameters caused by estimation errors, modeling errors, or linearization. For such perturbed systems, it is naturally required to design controllers to make sure that the controlled system remains stable in the presence of such uncertainties. Usually, the robust control problem for systems of DDEs has been handled by using Lyapunov functions, and employing linear matrix inequalities 共LMIs兲 or algebraic Riccati equations 共AREs兲 共see, e.g., Refs. 关4,5兴, and the references therein兲. Even though such approaches can be applied to quite general types of timedelay systems 共TDSs兲 共e.g., systems with multiple delays, timevarying delay兲, they provide only sufficient conditions and are substantially conservative because of their dependence on the selection of cost functions and their coefficients 关6兴. Moreover, general systematic procedures to construct appropriate Lyapunov functions are not available, and solving the resulting LMI/ARE can be nontrivial 关7兴. Analysis and design of control systems in the frequency domain is well established in control engineering. Stability is investigated based on the transfer function and the Nyquist criteria. By computing the robust stability margin in the Nyquist plane, the method has been used for robust control of systems of ODEs 关8兴 and also, DDEs 关9兴 with uncertainties. However, although being improved extensively, typically the method requires an exhaustive numerical search in the frequency domain plus an exhaustive search in the parameter domain 共see Table 1兲. On the other hand, while pursuing robust control, one may have to retain the positions of the eigenvalues to meet the response 1 Corresponding author. Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received July 1, 2008; final manuscript received February 12, 2010; published online April 14, 2010. Assoc. Editor: Rama K. Yedavalli.

specifications, such as time-domain specifications 共see Sec. 4兲, of the nominal system. For this reason, robust stability of the closedloop system has often been achieved through eigenvalue assignment combined with robust stability indices 共e.g., the stability radius concept in Sec. 3兲. Such indices set the upper limit on parameter perturbations and help select the positions of the desired eigenvalues in the complex plane. Then using an eigenvalue assignment method, it is possible to find feedback control for robust stabilization for systems of ODEs 共e.g., see Ref. 关10兴 and the references therein兲. However, systems of DDEs have an infinite number of eigenvalues, which are the roots of a transcendental equation, and it is not practically feasible to assign all of them. Thus, the usual pole placement design techniques for ODEs cannot be applied without considerable modification to systems of DDEs 关11兴. In this paper, we present a new approach to design robust controllers for a system of DDEs through eigenvalue assignment based on the Lambert W function approach. An eigenvalue assignment method for systems of DDEs was developed in Ref. 关2兴, based on a solution in terms of the Lambert W function 关12兴. Using that approach, summarized in Sec. 2 of this paper, one can design a linear feedback controller to place the rightmost eigenvalues at the desired positions in the complex plane and, thus, stabilize systems with a single time-delay. In that study, the critical rightmost subset of eigenvalues, which determines stability of the system, among the infinite eigenspectrum is assigned. This is possible because the eigenvalues are expressed in terms of the parameters of the system and each one is distinguished by a branch of the Lambert W function. The advantages of the Lambert W function-based approach over other existing methods 共e.g., rational approximations 关3兴, prediction-based approaches including finite spectrum assignment 共FSA兲 关15兴, continuous pole placement 关16兴, etc.兲 have been discussed in Ref. 关2兴 with examples. When uncertainty exists in the coefficients of the system, a robust control law, which can guarantee stability, is required. To realize robust stabilization, calculating allowable uncertainty 共i.e., the stability radius 关17兴, see Sec. 3.1兲, the rightmost eigenvalues are placed at an appropriate distance from the imaginary axis in order to guarantee stability. In this paper, the Lambert W function-based approach to eigenvalue assignment for DDEs in Ref. 关2兴 is combined with the stability radius concept to address the problem of robust stability of

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Table 1 Representative approaches for robust stability of systems of ODEs and DDEs Approach

Description

Ref. 共ODEs兲

Ref. 共DDEs兲

Lyapunov framework

Feasibility of LMI or solvability of ARE

Patel et al. 关13兴

Bengea et al. 关14兴

Nyquist method

Stability margin in the Nyquist plane

Postlethwaite and Foo 关8兴

Wang and Hu 关9兴

Find the positions of the desired eigenvalues via robust stability indices

Kawabata and Mori 关10兴

This paper using the Lambert W function

Eigenvalue assignment combined with robust stability indices

systems with uncertain parameters 共Sec. 3兲. Also, an approach for improvement of the transient response for systems of DDEs is presented. The method developed in Ref. 关2兴 also makes it possible to assign simultaneously the real and imaginary parts of a critical subset 共based on the concept of dominant poles 关18兴兲 of the eigenspectrum with linear feedback control. Therefore, guidelines, similar to those used for systems of ODEs to improve transient response, can be used for systems of DDEs via eigenvalue assignment by using the matrix Lambert W function-based approach to meet time-domain response specifications 共Sec. 4兲.

caused by time-delay term x共t − h兲, in the characteristic equation of DDEs. Thus, using the Lambert W function, the solution to Eq. 共1兲 was derived in Ref. 关12兴, and given by ⬁

x共t兲 =

兺e

k=−⬁

冕兺 t

Skt

CIk

+

⬁

eSk共t−␰兲CNk Bu共␰兲d␰

where 1 Sk = Wk共AdhQk兲 + A h

2 Eigenvalue Assignment Using the Lambert W Function Delay differential equations are a type of differential equation where the time derivatives at the current time depend on the solution, and possibly its derivatives, at previous times. A class of such equations that involve derivatives with delays as well as the solution itself has historically been called neutral DDEs 关19兴. In this paper, only retarded DDEs with a single delay where there is no time-delay in the derivative terms are considered. Consider a real linear time-invariant 共LTI兲 system of retarded DDEs with a single constant delay x˙ 共t兲 = Ax共t兲 + Adx共t − h兲 + Bu共t兲 x共t兲 = g共t兲, x共t兲 = x0,

t⬎0

t 苸 关− h,0兲 共1兲

t=0

where x共t兲 苸 R is a state vector; A 苸 R , Ad 苸 R , and B 苸 Rn⫻r are system matrices; and u共t兲 苸 Rr⫻1 is a function representing the external excitation. A specified preshape function g共t兲 and an initial point x0 are defined in the Banach space 关19兴. The existence and uniqueness of solutions to DDEs can be proven based on continuity 共i.e., g共0兲 = x0兲. However, even though g共0兲 ⫽ x0, the uniqueness and existence can be proven for a LTI system of DDEs with a single delay, as in Eq. 共1兲 关19兴. This section summarizes the solution to the system of DDEs in Eq. 共1兲 and the eigenvalue assignment method based on the solution form 关2兴. n

n⫻n

n⫻n

2.1 Solution in Terms of the Lambert W Function. The matrix Lambert W function Wk共Hk兲, which satisfies the definition 关20,21兴 Wk共Hk兲eWk共Hk兲 = Hk

共2兲

is complex valued, with a complex argument Hk, and has an infinite number of branches for k = −⬁ , . . . , −1, 0 , 1 , . . . , ⬁. The principal 共k = 0兲 and other 共k ⫽ 0兲 branches of the Lambert W function can be calculated analytically 关21兴, or using commands already embedded in the various commercial software packages, such as MATLAB, MAPLE, and MATHEMATICA. Due to its special definition in Eq. 共2兲, the Lambert W function has been known to be useful in handling the exponential term, 031003-2 / Vol. 132, MAY 2010

共3兲

0 k=−⬁

共4兲

The coefficient CIk in Eq. 共3兲 is a function of A, Ad, h, the preshape function g共t兲, and the initial condition x0, while CNk is a function of A, Ad, h, and does not depend on g共t兲 or x0. The numerical and analytical methods for computing CIk and CNk were developed in Refs. 关20,22兴. The following condition is used to solve for the unknown matrix Qk: Wk共AdhQk兲eWk共AdhQk兲+Ah = Adh

共5兲

The solution Qk for each branch k to Eq. 共5兲 is obtained numerically by using nonlinear solvers, for a variety of initial conditions, such as using the fsolve function in MATLAB. For a detailed study 共e.g., derivation and convergence of the solution兲, refer to Refs. 关12,23兴, and the references therein. Note that, compared with results by other existing methods for the series expansion of solution to DDEs, where eigenvalues are obtained from exhaustive numerical computation 共see, e.g., Refs. 关19,24,25兴, and the references therein兲, the solution in terms of the Lambert W function has a closed-form representation as an infinite series expressed in terms of the parameters of the DDE in Eq. 共1兲, i.e., A, Ad, and h. Hence, one can determine how those parameters are involved in the solution and, furthermore, how each parameter affects each eigenvalue and the solution. Also, each eigenvalue is distinguished in terms of k, which indicates the branch of the Lambert W function. For these reasons, the Lambert W function-based approaches have been applied to control problems and extended to other cases 共see, e.g., Refs. 关26–29兴兲. 2.2 Stability by the Principal Branch. The solution form in Eq. 共3兲 reveals that the stability condition for the system of Eq. 共1兲 depends on the eigenvalues of the matrix Sk, and, thus, also on the matrix eSk. A time-delay system characterized by Eq. 共1兲 is asymptotically stable if and only if all the eigenvalues of Sk, k = −⬁ , . . . , −1 , 0 , 1 , . . . , ⬁, have negative real parts. Computing the matrices Sk for an infinite number of branches is not practically feasible. However, if the coefficient matrix Ad does not have repeated zero eigenvalues, we have observed that the eigenvalues obtained using the principal branch 共k = 0兲 are the rightmost ones and determine the stability of the system 关30兴. Since there is currently no general proof, these observations are formally summarized here in the form of a Conjecture. That is 关30兴, Transactions of the ASME

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Conjecture: if Ad does not have repeated zero eigenvalues, then max兵R兵eigenvalues for the principal branch, k = 0其其 ⱖ R兵all other eigenvalues其

共6兲

Note that if Ad has repeated zero eigenvalues, the rightmost eigenvalues are obtained by using the principal branch 共k = 0兲, or k = ⫾ 1, for all cases considered. For the scalar case, it is proven that the root obtained using the principal branch always determines the stability of the system using monotinicity of the real part of the Lambert W function with respect to its branch, k 关31兴. Such a proof can readily be extended to systems of DDEs where A and Ad commute and, thus, are simultaneously triangularizable 关32兴. Although such a proof is not available in the case of general matrix-vector DDEs, we observe the same behavior in all the examples we have considered. We use that observation as the basis not only to determine the stability of systems of DDEs, but also to place a subset of the eigenspectrum at desired locations, as discussed in Sec. 2.3. In summary, to determine stability of Eq. 共1兲 one need to: 共1兲 solve Eq. 共5兲 for the principal branch 共k = 0兲 to find Q0, 共2兲 substitute Q0 into Eq. 共4兲 to get S0, and 共3兲 finally, check whether all the real parts of the eigenvalues of S0 are negative or not. 2.3 Eigenvalue Assignment. One of the main advantages of having a solution to the state equation in terms of system parameters is that by adjusting the parameters one can move the eigenvalues to desired positions in the complex plane. Through eigenvalue assignment, it is also possible to design feedback controllers. In Ref. 关2兴, the solutions in Eq. 共3兲 in terms of the matrix Lambert W function have also been used to develop an approach for eigenvalue assignment for systems of DDEs. In designing a linear feedback controller for a time-delay system, represented by DDEs in Eq. 共1兲, because there exists an infinite number of solution matrices Sk and the number of control parameters is finite, it is not feasible to assign all of them at once. Just by using the classical pole placement method for ODEs, placement of a selected finite number of eigenvalues may cause other uncontrolled eigenvalues to move to the right-half plane 共RHP兲 关16兴. However, the subsequent approach for control design using the matrix Lambert W function provides proper control laws without such loss of stability. A linear feedback containing current and delayed states has been used in literature 共see, e.g., Refs. 关2,14兴兲 to improve system response and is given by u共t兲 = Kx共t兲 + Kdx共t − h兲

共7兲

and Kd 苸 R are feedback gains. As mentioned where K 苸 R in Sec. 2.2, to design an effective control law for systems of DDEs is not as straightforward as for systems of ODEs, and various methods for feedback control has been studied extensively in the literature. For a detailed survey, refer to Refs. 关3,33兴, and the references therein. The system 共1兲 with the feedback control 共7兲 becomes a closedloop system r⫻n

r⫻n

x˙ 共t兲 = 兵A + BK其x共t兲 + 兵Ad + BKd其x共t − h兲

共8兲

The controllability of such system, using the solution form of Eq. 共3兲, was studied in Ref. 关34兴. The gains K and Kd are determined as follows. First, select desired eigenvalues ␭i,desired for i = 1 , . . . , n, and require that the selected eigenvalues become those of the matrix S0 ␭i共S0兲 = ␭i,desired

共9兲

for i = 1 , . . . , n, where ␭i共S0兲 is the ith eigenvalue of the matrix S0. Second, apply the new two closed-loop coefficient matrices A⬘ ⬅ A + BK and Ad⬘ ⬅ Ad + BKd, as in Eq. 共8兲 to Eq. 共5兲, and solve Journal of Dynamic Systems, Measurement, and Control

numerically to obtain the matrix Q0 for the principal branch 共k = 0兲. Note that K and Kd are unknown matrices with all unknown elements, and the matrix Q0 is a function of the unknown K and Kd. For the third step, substitute the matrix Q0 from Eq. 共5兲 into Eq. 共4兲 to obtain S0 and its eigenvalues as the function of the unknown matrix K and Kd. Finally, Eq. 共9兲 with the matrix S0 is solved for the unknown K and Kd using numerical methods, such as fsolve in MATLAB. Depending on the structure or parameters of a given system, there exist limitations on the rightmost eigenvalues and some values are not permissible. In that case, the above approach does not yield any solution for K and Kd. To resolve the problem, one can try again with fewer desired eigenvalues, or different values of the desired rightmost eigenvalues. Then, the solution K and Kd is obtained numerically for a variety of initial conditions by a trial and error procedure 关2兴. In summary, using the Lambert W function-based approach, the solution to systems of DDEs is given by Eq. 共3兲. With this solution method, a stability analysis can be conducted, as summarized in Sec. 2.2. Also, the solution and the stability analysis is used to assign the rightmost eigenvalues to desired locations in the complex plane, as summarized in Sec. 2.3. Thus, in subsequent sections, this approach can be used to design robust controllers through eigenvalue assignment combined with the stability radius concept, and to meet time-domain specifications based on the dominant pole concepts as in systems of ODEs. Besides the Lambert W function-based approach, various methods are available to stabilize time-delay systems without obtaining a solution to Eq. 共1兲. Approximating delay terms, using rational approximations 共including the Padé approximation兲 make timedelay systems finite dimensional in order to apply conventional control design methods. However, due to errors caused by approximation, the closed-loop system can become unstable 关3兴. Also, the Padé approximation in particular induces nonminimum phase zeros and, thus, high gain problems. The eigenvalues of the closed-loop system can be assigned at desired positions using the well-known Smith predictor 关35兴, where the delay is moved outside the feedback loop and the feedback controller can be designed in the same way as for ODEs 关36兴. However, in addition to mismatch problems, because of unstable pole-zero cancellation, the Smith predictor cannot be applied to unstable time-delay systems without modification 关37兴. To avoid such problems, an implementation method using numerical quadrature 共i.e., finite spectrum assignment兲 was suggested in Ref. 关15兴. However, because of the demand on computing the finite integral, this method also has limitations in implementation, and safe implementation is still an open problem 关3兴. The adaptive Smith controller can alleviate mismatch problems of the Smith predictor 关38兴, and has been implemented experimentally 关39兴, although its increasing computational complexity is an unavoidable limitation. The control problem for systems of DDEs has been handled using LMIs or AREs 共see, e.g., Refs. 关5,40,41兴, and the references therein兲. Even though such approaches can be applied to more general types of time delay systems 共e.g., systems with multiple delays, time-varying delay兲, they provide only sufficient conditions and are substantially conservative because of dependence on the selection of cost functions and their coefficients. In Ref. 关16兴, a numerical stabilization method was developed using a simulation package that computes the rightmost eigenvalues of the characteristic equation. For the obtained finite number of eigenvalues, the eigenvalues can be moved to the left-half plane 共LHP兲 using sensitivities with respect to changes in the feedback gain K. Compared with this approach, the matrix Lambert W function-based method yields the equation for assignment of the rightmost eigenvalues in terms of the parameters of the system. Using the algorithm, one can obtain the control gain to move the critical eigenvalues to the desired position without starting with their initial unstable positions or computing the rightmost eigenvalues and their sensitivities after every small movement in a quasicontinuous way. Using the Lambert W function, one can find the MAY 2010, Vol. 132 / 031003-3

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control gain independently of the path of the rightmost eigenvalues. Without planning the path, only from the desired values, the control gains for the system are obtained. This approach has been compared with the Lambert W function-based approach in Ref. 关2兴.

3

Robust Feedback

3.1 Stability Radius. To design robust feedback controllers through eigenvalue assignment, it is required to decide where is the appropriate positions in the complex plane to guarantee robust stability depending on the size of uncertainty. The decision can be made by using robust stability indices. The real stability radius, which is one of the indices and the norm of the minimum destabilizing perturbations, was obtained for linear systems of ODEs and a computable formula for the exact real stability radius was presented by Qiu et al. 关42兴. The real stability radius measures the ability of a system to preserve its stability under a certain class of real perturbations. The formula was extended to perturbed linear systems of DDEs in Ref. 关17兴. Assume that the perturbed system 共1兲 can be written in the form x˙ 共t兲 = 兵A + ␦A其x共t兲 + 兵Ad + ␦Ad其x共t − h兲 = 兵A + E⌬1F1其x共t兲 + 兵Ad + E⌬2F2其x共t − h兲

共10兲 li⫻n

, Fi 苸 R , and ⌬i 苸 R denotes the perturbawhere E 苸 R tion matrix. Provided that the unperturbed system 共1兲 is stable, the real structured stability radius of Eq. 共10兲 is defined as 关17兴 n⫻m

m⫻li

rR = inf兵␴1共⌬兲:system共10兲is unstable其

共11兲

where ⌬ = 关⌬1⌬2兴 and ␴1共⌬兲 denotes the largest singular value of ⌬. The largest singular value, ␴1共⌬兲 is equal to the operator norm of ⌬, which measures the size of ⌬ by how much it lengthens vectors in the worst case. Thus, the stability radius in Eq. 共11兲 represents the size of the smallest perturbations in parameters, which can cause instability of a system. And the real stability radius problem concerns the computation of the real stability radius when the nominal system is known. The stability radius is computed from 关17兴

再

rR = sup inf ␴2 ␻

␥苸共0,1兴

where ⍀共s兲 =

冉冋

R共⍀共j␻兲兲 − ␥I共⍀共j␻兲兲 ␥−1I共⍀共j␻兲兲 R共⍀共j␻兲兲

冋 册 F1

F2e−hs

册冊冎

共sI − A − Ad兲 E −1

共12兲

共13兲

In Eq. 共12兲, it is not practically feasible to compute the supremum value for the whole range of ␻ 苸 共−⬁ , ⬁兲. However, for the value ␻ⴱ, which satisfies

␻ⴱ ⬍ ¯␴共A兲 + ¯␴共Ad兲 + ¯␴共E兲␴គ 关W共0兲兴¯␴共关F1F2兴兲

W共0兲 =

冋册 F2

共− A − Ad兲 E −1

共15兲

Then, restab共␻ⴱ兲 ⱕ restab共␻兲 where restab共␻兲 =

再

inf ␴2

␥苸共0,1兴

冉冋

共16兲

R共⍀共j␻兲兲

− ␥I共⍀共j␻兲兲

␥ I共⍀共j␻兲兲

R共⍀共j␻兲兲

−1

册冊冎

共17兲

ⴱ

Therefore, one has only to check ␻ 苸 关0 , ␻ 兴 to obtain the supre031003-4 / Vol. 132, MAY 2010

3.2 Design of the Robust Feedback Controller. In this subsection, an algorithm is presented for the calculation of feedback gains to maintain stability for uncertain systems of DDEs. The approach to eigenvalue assignment using the Lambert W function is used to design robust linear feedback control laws, combined with the stability radius concept. The feedback controller can be designed to stabilize the nominal delayed system 共1兲 using the method outlined in Sec. 2.3 关2兴. However, if the system has uncertainties in the coefficients, which can be introduced by static perturbations of the parameters or can arise in estimating the parameters, the designed controller cannot guarantee stability. Therefore, a robust feedback controller is required when uncertainty exists in the parameters. Such a controller can be realized by providing sufficient margins in assigning the rightmost eigenvalues of the delayed system. However, conservative margins over those required can raise problems, such as cost of control. The stability radius, outlined in Sec. 3.1 provides a reasonable measurement of how large the margin should be. The basic idea of the proposed algorithm is to shift the rightmost eigenvalue to the left by computing the gains in the linear feedback controller 关2兴 and increase the stability radius until it becomes larger than the uncertainty of the coefficients. Then, one can obtain a robust controller to guarantee stability of the system with uncertainty. Algorithm 1. Designing a robust feedback controller for systems of DDEs with uncertainty. Step 1. Compute the radius r1 from actual uncertainties in parameters of given delayed system 共i.e., r1 = ␴1共⌬兲兲. Step 2. Using the eigenvalue assignment method presented in Sec. 2, compute K and Kd to stabilize the system. Step 3. Then, compute the theoretical stability radius of the stabilized system, r2 from Eq. 共12兲. Step 4. If r1 ⬎ r2, then the system can be destabilized by the uncertainties. Therefore, go to Step 2 and increase the margin 共compute K and Kd to move the rightmost eigenvalues more to the left兲. Example 1. From Ref. 关2兴, consider a system x˙ 共t兲 =

冋 册 冋 0 0 0 1

x共t兲 +

−1

−1

0

− 0.9

册

x共t − 0.1兲 +

冋册 0 1

u共t兲 共18兲

Without feedback control, the system in Eq. 共18兲 has one unstable eigenvalue 0.1098. Using feedback control as in Eq. 共7兲, designed by the method presented in Ref. 关2兴, if the desired rightmost eigenvalue is ⫺1, the computed gains are K = 关−0.1391 − 1.8982兴 and Kd = 关−0.1236 − 1.8128兴, and the stability radius in Eq. 共11兲 is 0.6255. However, if the system 共18兲 has uncertainties in the parameters

共14兲

where ¯␴共 · 兲 and ␴គ 共 · 兲 are the largest and smallest singular values of 共 · 兲, respectively, and F1

mum value in Eq. 共12兲. The obtained stability radius from Eq. 共12兲 provides a basis for assigning eigenvalues for robust stability of systems of DDEs with uncertain parameters.

x˙ 共t兲 =

再冋 册 冎 再冋 0 0 0 1

+

冋册 0 1

+ ␦A x共t兲 +

u共t兲

−1

−1

0

− 0.9

册 冎

+ ␦Ad x共t − 0.1兲 共19兲

and ␴共关␦A␦Ad兴兲 = 0.7, the system can become unstable due to uncertainty. To ensure stability, set the desired rightmost eigenvalue to be −2, then the computed gains are K = 关0.8805 − 2.1095兴 and Kd = 关0.9136 − 2.3932兴, and the stability radius in Eq. 共11兲 increases to 0.9151. Therefore, the system can remain stable despite the uncertainty 共␴共关␦A␦Ad兴兲 = 0.7兲. Table 1 shows the gains K and Kd corresponding to the several subsets of eigenvalues of S0. The computed stability radii versus the rightmost eigenvalues, moving from −0.5 to −4 are shown in Fig. 1. As seen in the figure, Transactions of the ASME

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1.1

Table 2 The gains K and Kd of the linear feedback controller in Eq. „7… corresponding to each of the rightmost eigenvalues. Computed by using the approach for eigenvalue assignment presented in Ref. †2‡.

1

K

0.8

−0.5 −1.0 −1.5 −2.0 −2.5 −3.0 −3.5 −4.0

0.7 0.6 0.5 0.4 0.3 0.2 −4

−3.5

−3

−2.5

−2

Rightmost Eigenvalues

−1.5

−1

−0.5

Fig. 1 As the eigenvalue moves left, then the stability radius increases consistently, which means improved robustness

for the system 共19兲, as the eigenvalue moves left, the stability radius increases monotonically. Note that, in general, an explicit relationship between the stability radius and the rightmost eigenvalues is not available, and moving the rightmost eigenvalues further to the left does not always lead to an increase in stability radius 关16兴. However, as shown above, by comparing the stability radius and uncertainty for a given system, Algorithm 1 can be used to achieve robust stability of TDSs with uncertainty. Michiels and Roose 关6兴, developed an algorithm to maximize the stability radius by calculating its sensitivity with respect to the feedback gain for a type of TDS. However, in maximizing the stability radius, the rightmost eigenvalues can be moved to undesired positions and one can fail to meet other specifications of the system response. If the system has relatively small uncertainty, instead of maximizing the stability radius, one can focus more on the position of eigenvalues to improve the transient response of the system, which will be discussed in a subsequent section. Also, robust stabilization of systems of DDEs has been investigated by conversion into rational discrete models 关43兴, or by canceling undesired dynamics of plants based on system model 关44兴. Compared with such methods, the Lambert W function-based approach presented here can improve accuracy and robustness of the controllers.

4

and and and and and and and and

关−0.6971 关−0.1391 关−0.3799 关0.8805 关1.8716 关2.5777 关2.8765 关3.1144

−6 −6 −6 −6 −6 −6 −6 −6

Kd 关−0.7098 关−0.1236 关1.0838 关0.9136 关0.8229 关0.7022 关0.9721 关1.1724

− 1.6893兴 − 1.8982兴 − 1.6949兴 − 2.1095兴 − 2.1103兴 − 1.7440兴 − 1.6818兴 − 1.5816兴

− 1.5381兴 − 1.8128兴 − 2.3932兴 − 2.3932兴 − 2.5904兴 − 2.9078兴 − 3.1311兴 − 3.3304兴

The eigenvalue is written as ␭ = ␴ ⫾ ␻di = −␨␻n ⫾ ␻n冑1 − ␨2i, the requirements for a step response are expressed in terms of the quantities, such as the rise time tr, the settling time ts, the overshoot M p, and the peak time t p. In the case of ODEs, if the system is second order without zeros, the quantities have exact representations tr =

1.8 , ␻n

ts =

4.6 , ␴

M p = e−␲␨/

冑1−␨2

,

tp =

␲ ␻d

共20兲

For all other systems, however, these provide only approximations, and can only provide a starting point for the design iteration based on the concept of dominant poles 关18兴. Figure 2 shows the responses corresponding to the rightmost eigenvalues considered in Table 3. Not surprisingly, the approximate values from Eq. 共20兲 in Table 4 are not exactly same as the results obtained from the responses in Fig. 2. But, for this example, the guidelines for second order ODEs still work well in the case of DDEs. That is, as raising the value of the imaginary part ␻d of the rightmost eigenvalue, the rise time tr of system, i.e., the speed at which the system respond to the reference input, decreases from 6.9 to 0.8. On the other hand, the maximum overshoot M p rises from 6% to 75%, which is typically not desirable. In this way, moving up or down the imaginary part, one can adjust the quantities related to the time-domain response and, thus, meet time-domain specifications. Figure 3 shows two responses corresponding to the several subsets of eigenvalues of S0, which have different real parts 共−0.2 and −0.5兲 with the same imaginary part 共⫾1.0i兲. As seen in the figure, the settling time, the rise time, and overshoot decrease with in-

Time-Domain Specifications

To meet design specifications in the time-domain, PID-based controllers have been combined with a graphical approach 关45兴, LQG method using ARE 关1兴, or Smith predictors 关46兴. These methods are available for systems with control delays. For systems with state delays, some sufficient conditions based on linear matrix inequality approaches have been proposed 共see, e.g., Ref. 关40兴 and the references therein兲. In this section, the Lambert W function-based approach, presented in Sec. 2, is applied to achieve time-domain specifications via eigenvalue assignment. Unlike other existing methods 共e.g., continuous pole placement in Ref. 关16兴兲, for the first time the Lambert W function-based approach can be used to assign the imaginary parts of system eigenvalues, as well as their real parts, for a critical subset of the infinite eigenspectrum. It is not practically feasible to assign the entire eigenspectrum; however, just by assigning some finite, but rightmost, eigenvalues the transient response of systems of DDEs can be improved to meet time-domain specifications for desired performance. Example 2. Consider the system in Eq. 共18兲. Table 2 shows the gains K and Kd corresponding to the several subsets of eigenvalues of S0, which have a real part, −0.2, and different imaginary parts, ⫾0.2i, ⫾0.5i, and ⫾1.0i. Journal of Dynamic Systems, Measurement, and Control

1.8 1.6

−0.2+1.0 i −0.2−1.0 i −0.2+0.5 i −0.2−0.5 i

1.4 1.2

Response, x(t)

Stability Radii, r

0.9

−0.2+0.2 i −0.2−0.2 i

1 0.8 0.6 0.4 0.2 0 0

5

10

15

Time, t

20

25

30

Fig. 2 Responses of the system in Eq. „18… with the feedback equation „7… corresponding to the rightmost eigenvalues in Table 3 with different imaginary parts of the rightmost eigenvalues

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Table 3 Gains K and Kd and parameters corresponding to several subsets of eigenvalues of S0 Rightmost eigenvalues

␴ ␻d ␻n ␨ = −␴ / ␻n K Kd

−0.2⫾ 0.2i

−0.2⫾ 0.5i

−0.2⫾ 1.0i

−0.5⫾ 1.0i

−0.2 0.2 0.2828 0.7 关0.0584 − 1.7867兴 关0.6789 2.3413兴

−0.2 0.5 0.5385 0.3714 关0.1405 − 1.7998兴 关0.7802 2.3204兴

−0.2 1.0 1.0198 0.1961 关0.4311 − 1.8152兴 关1.1421 2.2124兴

−0.5 1.0 1.1180 0.4472 关0.2380 − 2.1656兴 关0.9027 1.9451兴

Table 4 Comparison of the actual results and the approximations using Eq. „20… of timedomain specifications for Fig. 2 tr

ts

Mp

Rightmost eigenvalues

Approximate 共Eq. 共20兲兲

Actual

Approximate

Actual

Approximate

Actual

Approximate

Actual

−0.2⫾ 0.2i −0.2⫾ 0.5i −0.2⫾ 1.0i

6.4 3.3 1.8

6.9 2.5 0.8

23.0 23.0 23.0

23.0 23.0 27.0

4.6 28.5 53.4

6.0 31.0 75.0

15.7 6.3 3.1

14.6 6.0 2.4

creasing ␴, but the peak time remains almost the same. Thus, for this example, the guidelines based on the dominant poles concept for ODEs still work well in case of DDEs. The approach presented in this section is straightforward for systems of ODEs. However, it represents the first approach to assign the real and imaginary parts of the eigenvalues simultaneously to meet timedomain specifications for time-delay systems, and is very easy to use, since only the eigenvalues for the principal 共k = 0兲 branch are used. In this approach, we assign the real and imaginary parts of only the rightmost, thus dominant, eigenvalues. Even though the presented approach handles only a subset of eigenvalues from an infinite eigenspectrum, the subset is rightmost in the complex plane and dominates all other eigenvalues. Thus, for other LTI time-delay systems with a single delay this method can also be applied to achieve approximate time-domain specifications with linear feedback controllers. The approach presented follows the simple dominant poles design guidelines for ODEs, and provides an effective rule of thumb to improve the transient response of systems of DDEs. 1.8 1.6

−0.5+i −0.5−i

1.4 1.2

Response, x(t)

tp

1 0.8 0.6 0.4

−0.2+i −0.2−i

0.2

5

Conclusions and Future Work

In this paper, the eigenvalue assignment method, based on the Lambert W function, is applied to design linear robust feedback controllers and to meet time-domain specifications for LTI systems of DDEs with a single delay. An algorithm for design of feedback controllers to maintain stability for systems of DDEs with uncertainties in parameters is presented. With the algorithm, considering the size of the uncertainty in the coefficients of systems of DDEs via the stability radius, one can find appropriate gains of the linear feedback controller by assigning the rightmost eigenvalues. The procedure presented in this paper can be applied to uncertain systems, where uncertainty in the system parameters cannot be ignored. To improve the transient response of time-delay systems, the design guideline for systems of ODEs has been used via the Lambert W function-based eigenvalue assignment. The presented approach based on the concept of dominant poles is quite standard in case of ODEs. However, it has not been previously feasible to use such methods for systems of DDEs. Because, unlike ODEs, DDEs have an infinite number of eigenvalues, and controlling them has not been feasible due to the lack of an analytical solution form. Using the approach based upon the solution form in terms of the matrix Lambert W function, the analysis for robustness and transient response can be extended from ODEs to DDEs as presented in this paper. The presented method, which is directly related to the position of the rightmost eigenvalues, provides an accurate and effective approach to analyze stability robustness and transient response of DDEs. Even though it is not feasible to assign all of the infinite eigenvalues of time-delay systems, just by assigning the rightmost eigenvalues, one can control systems of DDEs in a way similar to systems of ODEs. This is the advantage of the Lambert W function-based approach over other existing methods.

Acknowledgment This work was supported by NSF Grant No. 0555765.

0 0

5

10

15

Time, t

20

25

30

References Fig. 3 Responses of the system in Eq. „18… with the feedback equation „7… corresponding to the rightmost eigenvalues in Table 3 with different real parts of the rightmost eigenvalues

031003-6 / Vol. 132, MAY 2010

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