Large Deviations in Linear Control Systems with Nonzero Initial

c Pleiades Publishing, Ltd., 2013. ISSN 0005-1179, Automation and Remote Control, 2013, Vol. 70, No. 6, pp. 1–21. ⃝ c B.T. Polyak, A.A. Tremba, M.V. Khlebnikov, P.S. Shcherbakov, G.V. Smirnov, 2013, Original Russian Text ⃝ published in Avtomatika i Telemekhanika, 2013, No. 6, XXX–YYY.

ADAPTIVE AND ROBUST SYSTEMS

Large Deviations in Linear Control Systems with Nonzero Initial Conditions B. T. Polyak∗,∗∗ , A. A. Tremba∗ , M. V. Khlebnikov∗ , P. S. Shcherbakov∗ , G. V. Smirnov∗∗∗ ∗

Trapeznikov Institute of Control Sciences, Russian Academy of Science, Moscow, Russia ∗∗ ITMO University, St. Petertsburg, Russia ∗∗∗ University of Minho, Braga, Portugal e-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Received MONTHEMBER XX, 2014

Abstract—Research in the transient response in linear systems with nonzero initial conditions was initiated by A.A. Feldbaum in his pioneering work [1] as early as in 1948. However later, studies in this direction have faded down, and since then, the notion of transient process basically means the response of the system with zero initial conditions to the unit step input. A breakthrough in this direction is associated with the paper [2] by R.N. Izmailov, where large deviations of the trajectories from the origin were shown to be unavoidable if the poles of the closed-loop system are shifted far to the left in the complex plane. In this paper we continue the analysis of this phenomenon for systems with nonzero initial conditions. Namely, we propose a more accurate estimate of the magnitude of the peak and show that the effect of large deviations may be observed for different root locations. We also present an upper bound on deviations by using the linear matrix inequality (LMI) technique. This same approach is then applied to the design of a stabilizing linear feedback aimed at diminishing deviations in the closed-loop system. Related problems are also discussed, e.g., such as analysis of the transient response of systems with zero initial conditions and exogenous disturbances in the form of either unit step function or harmonic signal.

1. INTRODUCTION Consider the stable linear system x˙ = Ax,

x(t) ∈ Rn ,

(1)

with nonzero initial condition x(0). It is of great interest to analyze the transient of the system, i.e. the behavior of its trajectory x(t) for all t ≥ 0.1 In particular, it is highly desirable to have estimates of the quantity ( ) |x(t)| ξ x(0) = max t>0 |x(0)| (here, | · | is a vector norm), i.e., the maximal deviation of the trajectory from the origin during the transient. If this quantity is large, we say that large deviations take place; in case these deviations are observed at the initial part of the trajectory, we refer to these as the peak effect. Needless to say, such characteristics of the transient regime are among the most important ones, having transparent engineering interpretations. Notably, nonzero initial conditions appear quite naturally in various situations, for instance in stabilization and control by means of observers [3] with unknown initial conditions; this is also 1

Clearly, by stability we have lim x(t) = 0. t→∞

1

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POLYAK, TREMBA, KHLEBNIKOV, SHCHERBAKOV, SMIRNOV

typical to switching systems [4] that reach a certain point in the phase space by a switch time instant. In Section 5 we show that the nature of peak effects (deviations) is very close to that of the well-studied phenomenon of overshoot in systems with zero initial conditions and unit step input. It is also worth mentioning that deviations of the solutions of linear ordinary differential equations with nonzero initial conditions are the subject of interest not only in control theory but also in the computational mathematics. Since the solution of system (1) has the form x(t) = eAt x(0), we have

|x(t)| . ξ = max max = max ∥eAt ∥, t>0 |x(0)|61 |x(0)| t>0

where ∥ · ∥ is the corresponding induced matrix norm, so that evaluation of the magnitude of deviation is directly related to the estimation of the matrix exponential.

Fig. 1. Typical behavior of the ∥eAt ∥ function.

In the celebrated paper [5] (also see its continuation [6]) on the methods of computing the matrix exponential, the existence of a noticeable maximum (often quite significant) of the function ∥eAt ∥ is considered typical (it is referred to as hump in the above mentioned works); see Fig. 1 borrowed from [6]. 4

x 10 10 8 6 4 2 0 0

10

20

30

40

50

60

Fig. 2. Plot of the matrix exponential in the stabilization problem for Boeing 767. AUTOMATION AND REMOTE CONTROL

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Several examples illustrating the effects of very large deviations in ”real-world” stabilization problems for control systems are also given in [6]; for instance, in the 55-dimensional stabilization problem for Boeing 767, the maximum of the quantity ∥eAt ∥ is of the order of 105 , see Fig. 2. However, in these works, no attention has been paid to the evaluation of the magnitude of the hump. A number of examples and results related to large deviations was obtained in [7–11]. In particular, for the nth-order matrix 



−1 −2 · · · · · · −2    0 −1 . . . . . . ...     . .  . . .  . . . . . . ..  B =  ..   .  ..  .  . −1 −2  . 0 · · · · · · 0 −1 the following estimate was shown to be valid, see [11]: ∥eBt ∥∞ −−−→ ∞ n→∞

∀t > 0.

In the present paper we generalize and unify the studies and results on large deviations and peaks in linear systems. The paper is organized as follows. In Section 2 we discuss the two cornerstone results known as Feldbaum’s theorem (1948) and Izmailov’s theorem (1987) on the connection between the root location of the system matrix and the characteristics of the transient. In Section 3.1, for various specific initial conditions, we provide precise estimates of the solutions of Eq. (1) with matrix represented in the companion form   

A=  

0 0 .. .

1 0 .. .

0 1 .. .

··· ··· .. .

0 0 .. .

−a0 −a1 −a2 · · · −an−1

     

(2)

and having equal real eigenvalues. The companion-form case is in the focus of our exposition, since it admits almost exhaustive analysis and may serve as the basis for the computation of lower bounds for systems of general form (see Section 3.5). The rest of Section 3 is devoted to lower estimates of the peak for various spectra of the system matrix. Specifically, estimates obtained in Section 2 are refined in Section 3.2; a similar result for small eigenvalues is established in Section 3.3; this subject is completed in Section 3.4 by discussing the behavior of trajectories for arbitrary root location (the problem is to estimate the minimal possible deviation as function of the order of the system). In Section 3.5, the presented results are extended toward general-form linear systems; the lower estimates thus obtained may however happen to be rather conservative. In the second part of the paper, Section 4, we discuss approaches to computing upper estimates of deviations. In particular, an LMI-based [12, 13] design procedure is proposed which guarantees “as small as possible” deviations; an upper estimate of the magnitude of the peak is also presented in Section 4. It is for a long time that the research in this direction is being performed, e.g., see [14–17]; we consider it natural to discuss lower and upper estimates in parallel. In Section 5 we discuss inherently similar effects of deviations in perturbed systems with zero initial conditions and their connection to the phenomena analyzed in the previous sections. Finally, open problems are discussed in the Conclusion. AUTOMATION AND REMOTE CONTROL

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2. THEOREMS BY FELDBAUM AND IZMAILOV The dependence of the peak effect on the spectrum (and other characteristics) of the matrix A was the subject of interest in the control community since the very beginning of studies in this area of research. One of the first works in this direction was the paper by A.A. Feldbaum [1] published in 1948. This paper dealt with SISO systems, which can be thought of as systems (1) with matrix of the form (2). The characteristic polynomial of such a matrix is seen to be equal to p(s) = sn + an−1 sn−1 + · · · + a0 .

(3)

Let λ1 , . . . , λn be its zeros, i.e. the eigenvalues of the matrix A, and let xi (t), i = 1, . . . , n, denote the ith component of the solution x(t) of system (1). A.A. Feldbaum obtained a number of results on the connection between the root location and the characteristics of the transient response. In particular, the following result was established. Proposition 1 (Feldbaum’s theorem). Assume that all zeros of the polynomial p(s) are real, except for possibly one complex conjugate pair, and their real parts are negative: Reλi 6 −σ < 0, and let the initial condition be

i = 1, . . . , n.

(

)⊤

x(0) = 1 0 · · · 0

.

Then the first component of the solution of system (1), (2) admits the estimate x1 (t) 6 y(t),

0 6 t < ∞,

where y(t) is the solution of the differential equation (s + σ)n y(t) = 0,

s=

d , dt

with initial conditions y(0) = 1, in other words,

y ′ (0) = · · · = y (n−1) (0) = 0; (

y(t) = v(tσ),

v(τ ) = 1 + τ +

τ2 τ n−1 ) −τ + ··· + e . 2 (n − 1)!

Note that the function y(t) is monotonically decreasing, so that for the considered initial conditions, there is no peak in the first component x1 (t) of the solution. It turns out however that for large values of n and σ, the last component experiences a very large peak.2 Moreover, for other initial conditions and complex eigenvalues, peak effects are unavoidable for the first component as well. The problem of large deviations was formulated by V.N. Polotskiy in a number of works published in the late 1970s; e.g., see [18, 19]. A fairly complete solution was obtained in 1987 by R.N. Izmailov [2]; the corresponding result is presented below. 2

Below, it will be shown that this result can be deduced from Proposition 1. AUTOMATION AND REMOTE CONTROL

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Proposition 2 (Izmailov’s theorem). System (1), (2) with Reλi 6 −σ < 0,

i = 1, . . . , n,

admits the following estimate: max max |x(t)| > cn σ n−1 ,

06t6 σ1 |x(0)|=1

(4)

where cn is a positive function that depends only on the order n of the system, and | · | is a norm in Rn . Clearly, the estimate (4) above implies the existence of large deviations for large values of n and σ > 1. Moreover, with Proposition 2, an important conclusion can be made on the inevitable peaks observed in the stabilization problems for systems with matrices of general form, see [2]. Finally, we note that an implicit estimate of the quantity cn was also presented in [2]. Proposition 3. Consider the following n-dimensional system with scalar control: x˙ = Ax + bu. For any stabilizing feedback u = k ⊤ x such that the eigenvalues of the closed-loop system satisfy the condition Reλi (A + bk ⊤ ) 6 −σ < 0, i = 1, . . . , n, there exists a constant C = C(A, b) > 0 which does not depend on k such that max max |x(t)| > Cσ n−1 .

06t6 σ1 |x(0)|=1

Therefore, a higher asymptotic decay rate of the process obtained by shifting the poles of the system far to the left in the complex plane is gained at the expense of large deviations at the initial part of the trajectory. These important results by Izmailov were somewhat generalized in [20], and a simpler proof was given in [21]. A new step forward in this direction was the proof of the fact that large deviations are typical also for small eigenvalues, see [9,10]; moreover, such effects are inherent to systems with other root locations. 3. LOWER ESTIMATES FOR VARIOUS ROOT LOCATIONS 3.1. Equal Eigenvalues: Exact Estimates Obviously, system (1) with matrix (2) is associated with the linear homogenous differential equation y (n) + a0 y (n−1) + · · · + an−2 y ′ + an−1 y = 0 (5) with initial conditions y(0) = x1 (0),

y ′ (0) = x2 (0),

...,

y (n−1) (0) = xn (0).

(6)

Clearly, the solution y(t) of Eq. (5), (6) is nothing but the first component x1 (t) of the solution of system (1), (2); respectively, the consecutive derivatives of y(t) are identically equal to the rest of the components: y (k) (t) ≡ xk+1 (t), k = 0, . . . , n − 1. AUTOMATION AND REMOTE CONTROL

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If the eigenvalues of the matrix A are all different, the solution of (5) is known to have the form y(t) = C1 eλ1 t + · · · + Cn eλn t , where the coefficients C1 , . . . , Cn , are defined by the initial conditions (6). If there exists an eigenvalue (say, λ1 ) of multiplicity l > 1, the solution is given by the quasipolynomial y(t) = (C1,1 + C1,2 t + · · · + C1, l tl−1 )eλ1 t + C2 eλ2 t + · · · + Cn−l eλn−l t , and the appearance of the polynomial multiplier is caused by the presence of the multiple root. It is this multiplier which is responsible for the transient for small values of t and, in particular, for the peak. This is the reason for paying special attention to the case where all eigenvalues of A coincide, i.e., the polynomial (3) is of the form p(s) = (s + σ)n . For certain initial conditions, the solutions of Eq. (5) can be easily obtained in closed form. In ( )⊤ particular, for x(0) = 1 0 · · · 0 we have (see Feldbaum’s theorem) (

y(t) = 1 + σt + · · · + (

)⊤

for x(0) = 0 0 · · · 1

we obtain y(t) =

(

σ n−1 tn−1 ) −σt e ; (n − 1)!

)⊤

and for x(0) = 1 1 · · · 1

tn−1 −σt e , (n − 1)!

the solution takes the form

(

y(t) = 1 + (1 + σ)t +

(1 + σ)n−1 n−1 ) −σt (1 + σ)2 2 t + ··· + t e . 2 (n − 1)!

The two latter solutions can be checked by direct substitution in the differential equation. For other initial conditions, the solution in the case of equal eigenvalues can be obtained by using the technique in [22]; in the general case, one has to resort to numerical methods. Using the closed-form solutions presented above, the estimates of deviations can be obtained. Theorem 1. Assume that all eigenvalues of the matrix (2) are equal to −σ < 0. Then the following is true for the solution x(t) of system (1), (2): a) for small σ we have max |x1 (t)| > cn σ 1−n , t> σ1

(

)⊤

for the initial conditions x(0) = 0 0 · · · 1

cn = and

max |x1 (t)| > cn σ −n , t> σ1

(

)⊤

for the initial conditions x(0) = 1 1 · · · 1

(n − 1)n−1 1−n e , (n − 1)!

1 cn = √ , e n

;

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b) for large σ and x(0) = 1 0 · · · 0

7

, the following inequality holds:

max |xn (t)| = cn σ n−1 ,

06t6 σ1

cn ≥

1 ; ne

c) for arbitrary values of σ, the following estimate is valid: max

max

t>0 |x(0)|∞ =1

|x(t)|∞ > κn ,

where the quantities κn > 0 can be evaluated numerically (see Table 2). The proof of this and the subsequent assertions are given in the Appendix. The table below presents the values of the respective constants3 appearing in Theorem 1; these are computed numerically for n ≤ 13. Table 1. Dependence of θn and cn on n from item b) in Theorem 1 n 2 3 4 5 6 7 θn 1 0.5858 0.4158 0.3225 0.2636 0.2228 cn 0.3679 0.2306 0.1682 0.1324 0.1092 0.09294 n 8 9 10 11 12 13 θn 0.1930 0.9037 0.8072 0.7295 0.6654 0.6118 cn 0.08088 0.07438 0.06945 0.06502 0.06106 0.05752

Here, θn denotes the quantity such that the maximum over t is attained with tn = θn /σ, and σn is the value of σ for which the lower bound is attained. Notably, the component x1 (t) takes large values for small σ (and large values of t), while the component xn (t) in the opposite situation. Table 2. Dependence of κn on n from item c) in Theorem 1 n σn κn n σn κn

2 2.7183 1.0453 8 1.0784 28.812

3 1.3508 1.5877 9 1.0763 52.684

4 1.1570 2.7716 10 1.0744 97.059

5 1.1053 4.9290 11 1.0725 180.05

6 7 1.0881 1.0816 8.8190 15.8811 12 13 1.0671 1.0616 341.33 653.11

Finally note that, since max

|x(0)|∞ =1

|x(t)|∞ =

max

|x(0)|∞ =1

|eAt x(0)|∞ = ∥eAt ∥∞ ,

item c) essentially deals with finding the ∞-norm of the matrix exponential. 3.2. Large Eigenvalues It turns out that “Izmailov-type” results (Proposition 2) can be derived from Feldbaum’s estimate (Proposition 1); namely, the following result is valid (for proof, see the Appendix). 3

All decimal digits reported in the table are correct; this equally relates to other tables accompanying the subsequent assertions. AUTOMATION AND REMOTE CONTROL

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Theorem 2. Under the conditions of Proposition 1 we have max |xn (t)| >

06t6 σ1

σ n−1 . ne

(7)

Hence, in contrast with the behavior of the first component (which, by Proposition 1, does not exceed the monotonically decreasing function y(t)), the last component for sure experiences peak at the initial part of the trajectory (for large values of σ). This phenomenon takes place for the specific initial condition ( )⊤ x(0) = 1 0 · · · 0 ; clearly, the maximum over all initial conditions bounded in the ∞-norm is no smaller. Importantly, the dependence of the peak magnitude on σ is the same as in Izmailov’s theorem, and the constant multiplier can be computed explicitly and has a very simple form.4 At the same time, the conditions of Theorem 2 are slightly more stringent, since they assume not arbitrary spectra, but rather those having no more than one complex conjugate pair. The result below was obtained in [10]; it refines the constant cn in Izmailov’s theorem and is valid for not necessarily real eigenvalues. Theorem 3. Assume that system (1), (2) with initial condition (

x(0) = 1 0 · · · 0)⊤ satisfies the requirement Reλi 6 −σ < 0, i = 1, . . . , n. Then the estimate ( log 2 ) > cn σ n−1 , cn = 2 log 2 − 1 ≈ 0.39 , x nρ ∞ n n is valid, where ρ = max |λi |. i=1,...,n

3.3. Small Eigenvalues The result below obtained in [10] relates to the case of eigenvalues located close to the imaginary axis. Theorem 4. Assume that system (1), (2) with initial condition (

x(0) = 0 0 · · · 1 satisfies the requirement Reλi < 0, i = 1, . . . , n. Then |x(γn /ρ)|∞ > cn ρ−(n−1) ,

)⊤

cn = γnn−1 (2 − eγn ),

where ρ = max |λi |, and γn > 0 is the solution of the equation i=1,...,n

2(n − 1) = (n − 1 + γ)eγ . Table 3 presents some numerical values of γn and cn as functions of n. Therefore, for ρ small, the deviations are large. Notably, in this case, the picture is totally different from that for large eigenvalues; specifically, the deviations are observed for large values of t, and it is the component x1 (t) that experiences peak, not its derivatives. 4

This constant can be refined by deriving a more accurate estimate in the proof of Theorem 2. AUTOMATION AND REMOTE CONTROL

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Table 3. Dependence of γn and cn on n in Theorem 4 n 2 3 4 5 6 7 8 9 10 γn 0.3748 0.4786 0.5304 0.5617 0.5829 0.5981 0.6096 0.6187 0.6259 cn 0.2044 0.0885 0.0448 0.0245 0.0140 0.0083 0.0050 0.0031 0.0019

3.4. Arbitrary Eigenvalues We briefly discuss the results on large deviations for arbitrarily located eigenvalues. A natural question is: What is the maximal possible value of κn > 1 such that the inequality max

max

t>0 |x(0)|∞ =1

|x(t)|∞ > κn

holds for any set of eigenvalues? It is reasonable to conjecture that the minimal (in the sense of the magnitude of κn ) deviation is attained with multiple eigenvalues λi ≡ −σ, i = 1, . . . , n. However this hypothesis is false; smaller deviations can be obtained by shifting some of the eigenvalues to the left.5 A meaningful result can seemingly be obtained by imposing bounds on the quantity ρ = max |λi | or, equivalently, on i=1,...,n

the coefficients of the characteristic polynomial. Example 1. Consider the third-order system of the form (1) with matrix 



0 1 0   0 0 1 A=  −5.9246 −12.2965 −11.1704 having eigenvalues λ1 = −10,

λ2,3 = −0.5852 ± j0.5.

The peak is equal to 1.4800, which is essentially less than κ3 = 1.5877 reported in Table 2. Similarly, for the fourth-order system with matrix 



0 1 0 0   0 0 1 0   A=    0 0 0 1 −0.9750 −27.7373 −3.0406 −14.8686 having eigenvalues λ1 = −14.7895,

λ2 = −0.0353,

λ3,4 = −0.0219 ± j1.3671,

the peak is equal to 2.5435, cf. κ4 = 2.7716. Both systems above were designed by minimizing numerically the value ξ(a1 , . . . , an−1 ) of the peak over the entries of the last row of A (equivalently, over the coefficients of the characteristic polynomial of the system). This was accomplished via use of the Matlab routine simulannealbnd which implements a popular simulated annealing algorithm targeted at solving global optimization problems. Hence, the question formulated at the beginning of this subsection remains open. 5

Essentially, this leads to the drop of degree of the characteristic polynomial of the system. AUTOMATION AND REMOTE CONTROL

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To conclude this subsection, we note that the situation gets much more complicated if more than two complex conjugate roots are allowed. The following instructive example is presented in the seminal paper [1] by A.A. Feldbaum. Consider the fourth-order system (1) with matrix (2) having eigenvalues λ1,2 = λ3,4 = −1 ± jβ (

)⊤

under the same initial condition x(0) = 1 0 · · · 0 . In that case, a closed-form solution of the system can be obtained, and the quantity max |x1 (t)| may happen to take arbitrarily large values t>0

for β large. Figure 3 depicts the plot of x1 (t) for β = 10. 2

1.5

1

x1(t)

0.5

0

−0.5

−1

−1.5

−2

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 3. Plot of x1 (t) for the pair of multiple roots λ1,2 = λ3,4 = −1 ± j10.

For the pair of multiple roots −1 ± j50 the result is even more impressive, see Fig. 4. 10

8

6

4

x1(t)

2

0

−2

−4

−6

−8

−10

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 4. Plot of x1 (t) for the pair of multiple roots λ1,2 = λ3,4 = −1 ± j50.

We also mention papers [7,8] where the analysis of the behavior of solutions for complex eigenvalues was performed; these however considered matrices which were not represented in the companion form (2). AUTOMATION AND REMOTE CONTROL

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3.5. Systems of General Type The lower estimates of deviations presented in the previous sections were obtained for systems with matrices in the companion form. These estimates can be used to evaluate deviations of systems in the general form; it is not quite clear how to find the corresponding estimates directly, without use of the results related to companion-form systems. Consider the system z˙ = Hz,

z(0) ̸= 0,

(8)

with general-form matrix H, which can be driven to the companion form by a linear transformation of the basis with a nonsingular matrix T ; e.g., see [13, 7]. By change of variables z = T −1 x in (8), we arrive at the system −1 x˙ = T {z } x | HT A

with companion-form matrix A = T HT −1 and initial condition x(0) = T z(0). Let m denote the smallest singular value of the matrix T −1 , i.e., |T −1 v| > m|v| for all v ∈ Rn , where | · | is the Euclidean vector norm. For the solutions z(t), x(t) of the corresponding systems we then have |z(t)| = |T −1 x(t)| > m|x(t)|, . |z(0)| = |T −1 x(0)| 6 ∥T −1 ∥ |x(0)| = M |x(0)|, where M denotes the largest singular value of T −1 . Hence, |z(t)| m |x(t)| > , |z(0)| M |x(0)| so that we can apply all lower estimates obtained in the previous sections for deviations of systems with companion-form matrix A. Of course, the estimates obtained in this way are rather conservative. For instance, systems with diagonal matrices H experience no peak at all; the quantity m/M is very small in this case, and the estimate thus obtained is trivial.

4. AN LMI-APPROACH TO THE MINIMIZATION OF DEVIATIONS In this section we make use of the linear matrix inequality (LMI) technique to construct quadratic Lyapunov functions and obtain upper bounds on deviations (the analysis problem). With the same apparatus, we will minimize deviations of closed-loop systems by means of properly chosen linear static state feedback (the design problem). Strictly speaking, this approach is not quite new; application of a similar machinery in [14–17] led to similar results, though the problem formulations in [14–17] slightly differ from ours. More importantly, it is believed that formulation of typical results on upper bounds in parallel with lower bounds is useful for better understanding of the deviation phenomenon. AUTOMATION AND REMOTE CONTROL

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4.1. Analysis: Upper Estimates We consider system (1) with stable matrix A ∈ Rn×n . It is well known that the stability of this system is equivalent to the existence of a quadratic Lyapunov function of the form V (x) = x⊤ Qx,

Q ≻ 0,

such that the matrix Q satisfies the linear matrix inequality A⊤ Q + QA ≺ 0, or, equivalently, the Lyapunov inequality AP + P A⊤ ≺ 0,

(9)

where it is denoted P = Q−1 ≻ 0. Consider now the ellipsoid {

E = x ∈ Rn :

x⊤ P −1 x 6 1

}

defined by the matrix P . Clearly, if an initial condition x(0) of system (1) belongs to this ellipsoid, the trajectory remains to stay in the ellipsoid for all time instants; this follows from the fact that the quadratic form x⊤ P −1 x is a Lyapunov function for this system. Therefore, if the ellipsoid E contains the unit ball B = {x ∈ Rn : |x|2 6 1}, then, for any initial condition in B, the trajectory will not leave the ellipsoid, and the following estimate for the Euclidean norm of the state vector x(t) is valid for all time instants: √ |x(t)|2 6 λmax (P ) = ∥P ∥2 . Since the condition B ⊆ E is equivalent to P < I, we arrive at the convex optimization problem ∥P ∥2 −→ min

subject to

AP + P A⊤ ≺ 0,

P < I,

(10)

with the matrix variable P = P ⊤ ∈ Rn×n . This problem is easy to solve in Matlab, e.g., by using the cvx toolbox [23]. In other words, we obtain the following result. Theorem 5. The solutions of system (1) with stable matrix A admit the estimate max max |x(t)|2 6

√

t>0 |x(0)|2 61

∥P ∥2 ,

where the matrix P is a solution of problem (10). Notably, in contrast to the setup in the previous sections, we seek to estimate the quantity max |x(t)|2 = max |eAt x(0)|2 = ∥eAt ∥2 ,

|x(0)|2 =1

|x(0)|2 =1

which is the spectral norm of the matrix exponential. In a number of publications (e.g., see [24, 25]), the Lyapunov equation A⊤ P + P A = −I with the identity matrix in the right-hand side is considered instead of the Lyapunov inequality (9); the resulting estimates are of higher conservatism. AUTOMATION AND REMOTE CONTROL

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Example 2. Consider system (1) with the companion matrix 



0 1 0   0 1 A= 0 −4 −6 −4 having eigenvalues λ1 = −2,

λ2,3 = −1 ± j.

Solving problem (10) gives λmax (P ) = 2.0930, while a straightforward optimization max max |eAt x(0)|2 = max ∥eAt ∥2 t>0 |x(0)|2 =1

t>0

results in the true peak value ξ = 1.6890. In other words, the conservatism of the estimate obtained via Theorem 5 is less than 20%. To compare, using the estimate √ max ∥eAt ∥2 6 C, C = ∥P −1 ∥2 ∥P ∥2 , A⊤ P + P A = −I, t>0

proposed in [24], we obtain Finally, the estimate

√ C = 4.7329, which is more than twice as worse as our result. max ∥eAt ∥2 6 t>0

√ C,

C=

( 1 + b )1+ 1 b

1 + ab

,

proposed in [25], where ⊤

λmax ( A+A 2 ) b= , κ results in

a=

κ ⊤ λmin ( A+A 2 )

,

κ=

1 , 2∥P ∥

A⊤ P + P A = −I,

√ C = 3.6827, which is half as worse as the LMI-estimate.

For systems whose eigenvalues are close to the boundary of the stability domain, the resulting LMI-estimate turns out to be arbitrarily accurate. Example 3. Consider system (1) with companion-form matrix A having eigenvalues λ1 = −0.1,

λ2,3 = −0.001 ± j10.

Solution of problem (10) yields λmax = 10.0499; on the other hand, computing max ∥eAt ∥2 , we t>0

obtain ξ = 10.0487, so that the ratio of these two quantities is just 0.9999. To conclude this subsection, consider the second-order system with the companion-form matrix (

A=

0 1 −1 −2

)

having eigenvalues λ1,2 = −1. This system experiences no peak at all, since the solution of problem (10) is given by the identity matrix P = I. AUTOMATION AND REMOTE CONTROL

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4.2. Design We now turn to the design problem and consider the control system x˙ = Ax + Bu

(11)

with state x(t) ∈ Rn , nonzero initial condition x(0), and control u ∈ Rm ; the pair (A, B) is assumed to be controllable. The goal is to find a stabilizing linear state feedback u = Kx

(12)

and the quadratic Lyapunov function for the closed-loop system such that its matrix is normminimal. In other words, by Theorem 5, this leads to the minimization of the estimate of deviations. With control (12), the closed-loop system (11) takes the form x˙ = (A + BK)x. Using the results in Section 4.1, we arrive at the minimization problem: ∥P ∥2 −→ min subject to the constraints (A + BK)P + P (A + BK)⊤ ≺ 0,

P < I.

After introducing an auxiliary matrix variable Y = KP , the first constraint takes the linear form AP + P A⊤ + BY + Y ⊤ B ⊤ ≺ 0, and we arrive at the convex minimization problem ∥P ∥2 −→ min

subject to

AP + P A⊤ + BY + Y ⊤ B ⊤ ≺ 0,

P < I,

(13)

with two matrix variables P = P ⊤ ∈ Rn×n and Y ∈ Rm×n . The final result is formulated below. Theorem 6. Let P , Y provide solutions to problem (13). Then the solutions of system (11) with regulator u = Kx, K = Y P −1 , admit the following estimate: max max |x(t)|2 6 χn , t>0 |x(0)|2 61

√

χn =

∥P ∥2 .

The following observation is important at this point. For system (11) in the companion form    A=  

0 0 .. .

1 0 .. .

0 1 .. .

··· ··· .. .

0 0 .. .

−a0 −a1 −a2 · · · −an−1



 

  ,  

0

  0  B=  ..  .

1

and controller with the gain matrix (

)

K = k0 k1 k2 · · · kn−1 , AUTOMATION AND REMOTE CONTROL

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the closed-loop system has the form    Ac =   

0 0 .. .

1 0 .. .

··· ··· .. .

0 1 .. .

0 0 .. .

k0 − a0 k1 − a1 k2 − a2 · · · kn−1 − an−1

   .  

Therefore, it is only the controller K that depends on the coefficients a0 , . . . , an−1 , while the matrix P and, respectively, the estimate χn are defined solely by the system dimension n. Moreover, the controller K has the form (

)

K = K0 + a0 a1 a2 · · · an−1 , where the controller K0 corresponds to the companion-form matrix A with zero last row. Table 4 presents the values of the upper bound χn in Theorem 6 as function of the dimension of the system. Note that for n = 2 there is no peak (cf. end of Section 4.1). Table 4. Dependence of χn on n in Theorem 6 n 2 3 4 5 6 7 8 9 10 χn 1 1.7321 2.4142 4.3357 6.7039 12.1266 19.6104 35.4971 58.8169

It is immediate to see that replacing the first constraint in problem (13) with AP + P A⊤ + BY + Y ⊤ B ⊤ 4 −2σP,

σ > 0,

ensures the stability of the matrix A + σI + BK. In other words, the degree of stability of the closed-loop system (11) is guaranteed to be no less than σ. It is interesting to compare the upper estimates χn with the lower ones, κn (see Table 2) obtained for the same values of σn . Note however that the quantities κn are associated with peaks measured in the ∞-norm, while χn relate to the Euclidean norm. The results of calculations are presented in Table 5. Table 5. Dependence of κn and χn on n for equal values of σn n 2 3 4 5 6 7 8 9 σn 2.7183 1.3508 1.1570 1.1053 1.0881 1.0816 1.0784 1.0763 κn 1.0453 1.5877 2.7716 4.9290 8.8190 15.881 28.812 52.684 χn 5.6147 9.5788 22.656 61.409 176.68 523.34 1571.1 4747.6

5. RELATED PROBLEMS In the previous sections we analyzed the effect of large deviations in disturbance-free systems, which was stipulated by nonzero initial conditions. We now turn to systems with zero initial conditions but having a nonzero input, and show that they may exhibit a similar behavior. 5.1. Unit Step Input Consider a stable system with zero initial conditions x˙ = Ax + bu, AUTOMATION AND REMOTE CONTROL

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and scalar unit-step input u(t) ≡ 1, The solution has the form

t > 0.

x(t) = −A−1 b + eAt A−1 b,

where x(t) −−−→ x ¯, t→∞

−A−1 b

and x ¯= is the steady-state value of the output variable. Introducing the error e(t) = x − x ¯ = eAt A−1 b, we see that estimation of the maximum value of this quantity6 is nothing but the evaluation of deviation of system (1) with fixed initial conditions x(0) = A−1 b. Let us demonstrate that the error may be large for specific values of the vector b. To this end, in accordance with Proposition 3, we pick the initial condition |x(0)| = 1 of the unperturbed system (14) in such a way as to maximize the peak of its trajectory x(t) = eAt x(0). Then the maximum of |e(t)| is attained with b = Ax(0) and is equal to max |e(t)| = max max |eAt x(0)| = max ∥eAt ∥. t>0

t>0 |x(0)|=1

t>0

We this arrived at exactly the problem of estimation of large deviations in disturbance-free systems with nonzero initial conditions. 5.2. Harmonic Input Let us now consider the stable system (14) with the complex-valued (for simplicity of calculations) harmonic input signal u(t) = ejωt (15) with frequency ω > 0. As it is well known from the classical regulation theory, the steady-state solution of this system is also represented by a harmonic function; however, the transient response may experience large deviations from the steady-state regime at the initial part of the trajectory. This phenomenon is in complete analogy with the effects typical to the problems considered in the previous sections. Indeed, by straightforward calculations we obtain the following expression for the solution of the system: ∫t

x(t) =

. eA(t−τ ) bejωτ dτ = H(jω)ejωt b − H(jω)eAt b = x ¯(t) − xd (t),

0

where H(jω) = (jωI − A)−1 is the matrix frequency response of the system, while x ¯(t) and xd (t) denote the steady-state harmonic value of the state vector and its decaying component, respectively. The behavior of the quantity xd (t) = H(jω)eAt b is also seen to be defined by the matrix exponential, and the vector b may be thought of as a nonzero initial condition. The only difference with the disturbance-free system is the presence of the matrix multiplier H(jω). 6

Essentially, this problem is equivalent to the classical overshoot problem in linear systems. AUTOMATION AND REMOTE CONTROL

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Example 4. Consider system (14) with the vector (

)⊤

b = 0 ··· 0 1

.

We will be interested in the behavior of the real part of the first component x1 (t) of the solution when the input is of the harmonic form (15)7 . Specifically, we consider the maximum excess max |Rex1 (t)| t>0

max |Re¯ x1 (t)| t>0

of the magnitude of the output over the amplitude of the steady-state signal. 8

6

4

2

0

−2

−4

−6

−8

−10

−12

0

10

20

30

40

50

60

70

Fig. 5. Steady state and the transient in the system with harmonic input.

Figure 5 depicts the plots of the output variable Rex1 (t) (dashed line) and its steady-state value Re¯ x1 (t) (solid line) for the 10th-order system (14) in the canonical controllable form with equal eigenvalues λi ≡ −0.5 and harmonic input (15) with ω = 0.9. The amplitude of the steady-state signal is equal to max |Re¯ x1 (t)| = 0.7473, t>0

while the maximum value of the output is max |Rex1 (t)| = 11.5405, t>0

which is more than 15 times as large as the steady-state value. This difference gets much bigger as the frequency ω grows (more than three thousand times for ω = 2) and as the eigenvalues λi approach the imaginary axis (about fifteen hundred times for λi ≡ −0.25). To conclude this section, we mention a similar result presented in the paper [26] by N.G. Chebotarev. Specifically, the maximum value of the output of any stable linear system with real eigenvalues λi and bounded input f (t) admits the following upper estimate: sup |f (t)| max |y(t)| 6 t>0

t>0

∏ i

7

|λi |

.

Under these conditions, the steady-state harmonic output is given by |h1n |, the absolute value of the (1, n)th entry of the matrix frequency response H(jω). AUTOMATION AND REMOTE CONTROL

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POLYAK, TREMBA, KHLEBNIKOV, SHCHERBAKOV, SMIRNOV

As the roots λi approach zero, the right-hand side of the inequality is seen to experience unbounded growth. 6. DISCUSSION AND OPEN PROBLEMS Estimation of deviations in the transient regimes under nonzero initial conditions renders importance, yet being not analyzed in full. The paper presents several results in this direction; however numerous problems remain to be open. Below is a list of some problems that admit simple formulations; we believe these will attract attention of researchers. 1. Formulation of more accurate upper and lower bounds on deviations for various root locations; these bounds are to be of the same order of magnitude. 2. Finding lower bounds on deviations for systems with companion form matrices (2) under the only assumption that the spectral radius of A is bounded (see Section 3.4). 3. Direct computation of lower bounds of deviations for classes of matrices different from companion form. Certain results for Jordan-form matrices are obtained in [7, 8]. 4. Formulation of the results in Section 3 (lower bounds in the analysis problem) for discrete time systems. There exist some results in this field, e.g., those related to upper estimates of the norm of the matrix exponential maxk ∥Ak ∥ for Schur stable matrices. Note also that upper bounds in the design problem can be immediately obtained by using the discrete-time Lyapunov equation, see [27]. ACKNOWLEDGEMENTS This work was supported by the Mega-Grant of the Russian Federation (project 14.Z50.31.0031), the Russian Foundation for Basic Research (projects Nos. 14-07-00067-a and 14-08-01230-a), and Portugal grants FCT, COMPETE, QREN, FEDER, Project VARIANT (PTDC/MAT/111809/2009). The authors are thankful to N.A. Polinova, J. Whidborne, M.M. Kogan, and V.F. Sokolov for providing us with useful bibliographical references, critical comments, and discussion of the results. APPENDIX Proof of Theorem 1. a) For the initial condition

(

x(0) = 0 0 · · · 1 we have y(t) =

)⊤

tn−1 −σt e . (n − 1)!

This function attains its maximum value ymax =

(n − 1)n−1 1−n 1−n σ e = O(σ 1−n ) (n − 1)!

at the time instant t = (n − 1)/σ. (

)⊤

For x(0) = 1 1 · · · 1

we have

(

y(t) = 1 + (1 + σ)t +

(1 + σ)2 2 (1 + σ)n−1 n−1 ) −σt t + ··· + t e , 2 (n − 1)!

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hence,

(

y(n/σ) > 1 +

19

1 )n ( n )n 1 1 > √ n, σ e n! e nσ

so that for small values of σ we obtain |y(n/σ)| = O(σ −n ). b) For the initial condition

(

x(0) = 1 0 · · · 0 we have y(t) =

)⊤

n−1 ∑

(σt)k −σt e , k! k=0

whence it follows that

σ n tn−1 −σt e . (n − 1)! Recalling the definition of generalized Laguerre polynomials y ′ (t) = −

(α)

Lk (θ) =

θ−α eθ dk k+α −θ (θ e ), k! dθk

we obtain (1)

y (n−1) (t) = −Ln−2 (σt) so that

(n−1) ( θn ) y = cn σ n−1 ,

σ

where θn =

σ n te−σt , n−1

|θn e−θn Ln−2 (θn )| , n−1 (1)

cn =

(1) arg max |θe−θ Ln−2 (θ)|. θ>0

To substantiate a simpler estimate cn > 1/(ne), it suffices to note that y(t) = v(τ ), τ = σt, where v(τ ) is given in Proposition 1, and repeat the proof of Theorem 2. c) The results presented in Table 2 are obtained via numerical minimization of the quantity max |x(t)|∞ over initial conditions |x(0)|∞ = 1 and σ > 0. t>0

Proof of Theorem 2. First note that v(τ ) < 1, so that (τn

)

τ n −τ e . n! n! Next, by expanding the function x1 (t) into the Taylor series, we obtain 1 − v(τ ) =

+ · · · e−τ >

(A.1)

(0) ( τ )n−2 x1 x (θτ /σ) ( τ )n−1 + ··· + 1 x1 (τ /σ) = x1 (0) + + = σ (n − 2)! σ (n − 1)! σ xn−1 (0) ( τ )n−2 xn (θτ /σ) ( τ )n−1 τ = = x1 (0) + x2 (0) + · · · + + | {z } σ (n − 2)! σ (n − 1)! σ (n−2)

τ x′1 (0)

=1

|

(n−1)

{z

}

=0

=1+

xn (θτ /σ) ( τ )n−1 , (n − 1)! σ

0 6 θ 6 1.

Accounting for (A.1) and the Feldbaum theorem, we find |xn (θτ /σ)| =

σ n−1 (n − 1)! 1 − x1 (τ /σ) > n−1 τ ) σ n−1 (n − 1)! τ n −τ τ σ n−1 (n − 1)! ( 1 − v(τ ) > e = σ n−1 e−τ . > n−1 n−1 τ τ n! n

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Finally, by letting τ = 1, we arrive at the desired estimate (7) for t = θ/σ ∈ [0, 1/σ]. The theorem is proved. REFERENCES 1. Feldbaum, A.A., On the Root Location of Characteristic Equations of Control Systems, Avtom. Telemekh., 1948, no. 4, pp. 253–279. 2. Izmailov, R.N., The “Peak” Effect in Stationary Linear Systems with Scalar Inputs and Outputs, Autom. Remote Control, 1987, vol. 48, no. 8, pp. 1018–1024. 3. Luenberger, D.G., An Introduction to Observers, IEEE Trans. Autom. Control, 1971, vol. 35, pp 596– 602. 4. Liberzon, D., Switching in Systems and Control, Boston: Birkh¨auser, 2003. 5. Moler, C. and Van Loan, C., Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review, 1978, vol. 20, pp. 801–836. 6. Moler, C. and Van Loan, C., Nineteen Dubious Ways to Compute the Exponential of a Matrix, TwentyFive Years Later, SIAM Review, 2003 vol. 45, no. 1, pp. 3–49. 7. Akunov, T.A., Dudarenko, N.A., Polinova, N.A., et al., Analysis of Processes in Continuous Time Systems with Multiple Complex Conjugate Eigenvalues of the State Matrices, Nauchn.-tekhn. Vestn. Inform. Technologii, Mech., Optiki, 2013, no. 4(86), pp. 25–33. 8. Akunov, T.A., Dudarenko, N.A., Polinova, N.A., et al., Degree of Proximity of Simple and Multiple Eigenvalue Structures: Minimization of the Trajectory Peaks in Unperturbed Motion of Aperiodic Systems, Nauchn.-tekhn. Vestn. Inform. Technologii, Mech., Optiki, 2014, no. 2(90), pp. 39–46. 9. Smirnov, G., Bushenkov, V., and Miranda, F., Advances on the Transient Growth Quantification in Linear Control Dystems, Int. J. Appl. Math. Statist., 2009, vol. 14, pp. 82–92. 10. Polyak, B.T. and Smirnov, G.V., Large Deviations in Continuous-Time Linear Single-Input Control Systems, Proc. 19 IFAC World Congr., Cape Town, Aug. 2014, pp. 5586–5591. 11. van Dorsselaer, J.L.M., Kraaijevanger, J.F.B.M., and Spijker, M.N., Linear Stability Analysis in the Numerical Solution of Initial Value Problems, Acta Numerica, 1993, vol. 2, pp. 199–237. 12. Boyd, S., El Ghaoui, L., Feron, E., et al., Linear Matrix Inequalities in Systems and Control Theory, Philadelphia: SIAM, 1994. 13. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems Subject to Exogenous Disturbances: The Linear Matrix Inequalitiy Technique), Moscow: LENAND, 2014. 14. Hinrichsen, D., Plischke, E., and Wurth, F., State Feedback Stabilization with Guaranteed Transient Bounds, Proc. 15 Int. Symp. Math. Theory Networks & Syst. (MTNS-2002), Notre Dame, Indiana, Aug. 2002, paper no. 2132 (CDROM). 15. Whidborne, J.F. and McKernan, J., On Minimizing Maximum Transient Energy Growth, IEEE Trans. Autom. Control, 2007, vol. 52, no. 9, pp. 1762–1767. 16. Balandin, L.V. and Kogan, M.M., Lyapunov Function Method for Control Law Synthesis under One Integral and Several Phase Constraints, Differential Equations, 2009, vol. 45. no. 5, pp. 670–679. 17. Whidborne, J.F. and Amar, N., Computing the Maximum Transient Energy Growth, BIT Numerical Math., 2011, vol. 51, no. 2, pp. 447–557. 18. Polotskiy, V.N., On the Maximal Errors of an Asymptotic State Identifier, Autom. Remote Control, 1978, vol. 39, no. 8, pp. 1116–1121. 19. Polotskiy, V.N., Estimation of the State of Single-Output Linear Systems by Means of Observers, Autom. Remote Control, 1980, vol. 41, no. 12, pp. 1640–1648. AUTOMATION AND REMOTE CONTROL

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20. Sussmann, H.J. and Kokotovic, P.V., The Peaking Phenomenon and the Global Stabilization of Nonlinear Systems, IEEE Trans. Autom. Control., 1991, vol. 36, no. 4, pp. 424–439. 21. Bushenkov, V. and Smirnov, G., Stabilization Problems with Constraints: Analysis and Computational Aspects, Amsterdam: Gordon and Breach, 1997. 22. Polyak, B.T. and Tremba, A.A., Closed-Form Solution of Linear Differential Equations with Equal Roots of the Characteristic Equation Proc. XII All-Russia Workshop on Control Problems (VSPU2014), Moscow, Jun. 2014, pp. 212–217. 23. Grant, M. and Boyd, S., CVX: Matlab Software for Disciplined Convex Programming (web page and software), URL http://stanford.edu/boyd/cvx. 24. Bulgakov, A.Ja., An Efficiently Calculable Parameter for the Stability Property of a System of Linear Differential Equations with Constant Coefficients, Siberian Math. J., 1980, vol. 21, no. 3, pp. 339-347. 25. Nechepurenko, Yu.M., Bounds for the Matrix Exponential Based on the Lyapunov Equation and Limits of the Hausdorff Set, Comput. Math. Math. Phys., 2002, vol. 42, no. 2, pp. 125-134. 26. A Letter by N.G. Chebotarev on a Mathematical Problem Related to the Evaluation of the Regulated Coordinate when the Disturbing Force is Bounded in Absolute Value, Avtom. Telemekh., 1948, vol. IX, no. 4, pp. 331–334. 27. Kogan, M.M. and Krivdina, L.N., Synthesis of Multipurpose Linear Control Laws of Discrete Objects under Integral and Phase Constraints, Autom. Remote Control, 2011, vol. 72, no. 7, pp. 1427–1439.

This paper was recommended for publication by A.P. Kurdyukov, a member of the Editorial Board

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