A Unified Approach to High-Gain Adaptive Controllers

Abstract

It has been known for some time that proportional output feedback will stabilize MIMO, minimum-phase, linear time-invariant systems if the feedback gain is sufficiently large. High-gain adaptive controllers achieve stability by automatically driving up the feedback gain monotonically. More recently, it was demonstrated that sample-and-hold implementations of the high-gain adaptive controller also require adaptation of the sampling rate. In this paper, we use recent advances in the mathematical field of dynamic equations on time scales to unify and generalize the discrete and continuous versions of the high-gain adaptive controller. We prove the stability of high-gain adaptive controllers on a wide class of time scales.

1. Introduction

The concept of high-gain adaptive feedback arose from a desire to stabilize certain classes of linear continuous systems without the need to explicitly identify the unknown system parameters. This type of adaptive controller does not identify system parameters at all, but rather adapts the feedback gain itself in order to regulate the system. A number of papers examine the details of various kinds of high-gain adaptive controllers [1–4], among others. More recently, several papers have discussed one particularly practical angle on the high-gain adaptive controller, namely, how to cope with input/output sampling. In particular, Owens [5] showed that it is not generally possible to stabilize a linear system with adaptive high-gain feedback under uniform sampling. Thus, Owens et al., develop a mechanism to adapt the sampling rate as well as the gain, a notion subsequently improved upon by Ilchmann and Townley [6, 7], Ilchmann and Ryan [8], and Logemann [9].

In this paper, we employ results from the burgeoning new field of mathematics called dynamic equations on time scales to accomplish three principal objectives. First, we use time scales to unify the continuous and discrete versions of the high-gain controller, which have previously been treated separately. Next we give an upper bound on the system graininess to guarantee stabilizability for a much wider class of time scales than previously known, including mixed continuous/discrete time scales. Third, the paper represents the first application of several fairly recent advances in stability theory and Lyapunov theory for systems on time scales, and two new lemmas are presented in that vein. We also give a simulation of a high-gain controller on a mixed time scale.

2. Background

We first state two assumptions that are required in the subsequent text.

(A1)The system model and feedback law are given by the linear, time-invariant, minimum phase system 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑦(𝑡)=𝐶𝑥(𝑡),𝑥(0)=𝑥₀,𝑢(𝑡)=−𝑘(𝑡)𝑦(𝑡) (2.1)(2.2) for 𝑡≥0. System parameters 𝐴∈ℝ^𝑛×𝑛, 𝐵,𝐶^𝑇∈ℝ^𝑛×𝑚, 𝑥₀∈ℝ^𝑛, and 𝑛 are unknown. The feedback gain 𝑘∶ℝ→ℝ⁺ is piecewise continuous, and nondecreasing as 𝑡→∞. By minimum phase, we mean that the polynomial det𝐴−𝜆𝐼𝐵𝐶0 (2.3) with 𝜆∈ℂ is Hurwitz (zeros in open left hand plane).(A2)Furthermore, (𝐶𝐵)^𝑇+(𝐶𝐵)>0, (2.4) that is, (𝐶𝐵)^𝑇+(𝐶𝐵) is positive definite. (In [3], it is pointed out that a nonsingular input/output transformation 𝑇 always exists such that 𝐵=𝐵𝑇⁻¹ and 𝐶=𝑇𝐶 give (𝐶𝐵)^𝑇+(𝐶𝐵)>0.)

Under these conditions, it has been known for some time (e.g., [1]) that there are a wide class of gain adaptation laws 𝑘(𝑡)=𝑓(𝑦(𝑡)), 𝑓∶ℝ^𝑚→ℝ, that can asymptotically stabilize system (2.1) in the sense that

𝑦∈𝐿²[𝑡₀,∞),lim_𝑡→∞𝑘(𝑡)<∞. (2.5) Subsequently, various authors [1, 5, 6, 10] assumed that the output is obtained via sample-and-hold, that is, 𝑦_𝑖∶=𝑦(𝑡_𝑖) with 𝑘_𝑖∶=𝑘(𝑡_𝑖) and 𝑖∈ℕ₀. Thus it becomes necessary also to adapt the sample period ℎ_𝑖∶=𝑡_𝑖+1−𝑡_𝑖, so the closed-loop control objectives are

𝑦_𝑖∈ℓ²[𝑖₀,∞),lim_𝑖→∞𝑘_𝑖<∞,lim_𝑖→∞ℎ_𝑖>0. (2.6) Though several variations on these results exist, these remain the basic control results for continuous and discrete high-gain adaptive controllers. The continuous and discrete cases have previously been treated quite differently, but we now construct a common framework for both using time-scale theory.

3. A Time-Scale Model

The system of (A1) can be replaced by 𝑥^Δ(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑢(𝑡),𝑦(𝑡)=𝐶𝑥(𝑡),𝑥(0)=𝑥₀,𝑢(𝑡)=𝑘(𝑡)𝑦(𝑡),𝑡∈𝕋, (3.1) where 𝕋 is any time scale unbounded above with 0∈𝕋. With a series expansion similar to [1], we see that

𝐴(𝑡)∶=expc(𝜇(𝑡)𝐴)𝐴,𝐵(𝑡)=expc(𝜇(𝑡)𝐴)𝐵, (3.2) where expc is the matrix power series function

1expc(𝑋)∶=𝐼+21𝑋+6𝑋²1+⋯+𝑋𝑛!^𝑛−1+⋯ (3.3) and 𝜇 is the time scale graininess. Implementing control law, (2.2) then gives

𝑥^Δ(𝑡)=𝒜(𝑡)𝑥(𝑡),𝒜(𝑡)∶=expc(𝜇(𝑡)𝐴)(𝐴−𝑘(𝑡)𝐵𝐶). (3.4) Note that 𝐴,𝐵, and 𝒜 may all be time-varying, but we will henceforth drop the explicit reference to 𝑡 for these variables. For future reference, we also note that if sup_𝑡∈𝕋𝜇(𝑡)<∞, then ‖𝐴‖ and ‖𝐵‖ are also bounded (c.f. Appendix).

The design objectives are to find graininess 𝜇 and feedback gain 𝑘 as functions of the output 𝑦, 𝜇(𝑡)=𝑔(𝑦(𝑡)),𝑔∶ℝ^𝑚→[0,∞),𝑘(𝑡)=𝑓(𝑦(𝑡)),𝑓∶ℝ^𝑚→ℝ⁺ (3.5) with 𝑡∈𝕋 and 𝑘∈𝐶_rd(𝕋) nondecreasing, such that

𝑦∈𝐿²[𝑡₀,∞)_𝕋∶={𝑤∶^∞_𝑡0|𝑤(𝑡)|²Δ𝑡<∞},lim_𝑡→∞𝑘(𝑡)<∞. (3.6)

It is important to keep in mind the generality of the expressions above. A great deal of mathematical machinery supports the existence of delta derivatives on arbitrary times scales, as well as the existence and characteristics of solutions to (3.4). See, for example, [11, 12].

4. Stability Preliminaries

We begin this section with a definition and theorem from the work of Pötzsche et al. [13].

Definition 4.1. The set of exponential stability for the time-varying scalar equation 𝑧^Δ(𝑡)=𝜆(𝑡)𝑧(𝑡), 𝑧₀∶=𝑧(𝑡₀) with 𝜆∶𝕋→ℂ and 𝑧,𝜆∈𝐶_rd(𝕋), is given by 𝒮(𝕋)∶=𝒮_ℂ(𝕋)∪𝒮_ℝ(𝕋), (4.1) where 𝒮_ℂ(𝕋)∶={𝜂(𝑡)∈ℂ∶𝛼=−limsup_𝑡→∞1𝑡−𝑡₀^𝑡_𝑡0log|1+𝜇(𝑡)𝜂(𝑡)|𝒮𝜇(𝑡)Δ𝑡>0},_ℝ(𝕋)∶={𝜂(𝑡)∈ℝ∶∀𝑡∈𝕋,∃𝜏>𝑡with𝜏∈𝕋suchthat1+𝜇(𝑡)𝜂(𝑡)=0}, (4.2)(4.3) with 𝜂∈𝐶_rd(𝕋) and arbitrary 𝑡₀∈𝕋.

Theorem 4.2 (see [13]). Solutions of the scalar equation 𝑧^Δ(𝑡)=𝜆(𝑡)𝑧(𝑡) are exponentially stable on an arbitrary 𝕋 if and only if 𝜆(𝑡)∈𝒮(𝕋).

We note here that Pötzche, Siegmund, and Wirth did not explicitly consider scenarios where 𝜂 is time-varying, but their stability analysis remains unchanged for 𝜂(𝑡).

The set 𝒮_ℝ(𝕋) contains nonregressive eigenvalues 𝜆(𝑡), and a loose interpretation of 𝒮_ℂ(𝕋) suggests that it is necessary for a regressive eigenvalue to reside in the area of the complex plane where ‖1+𝜇(𝑡)𝜆(𝑡)‖<1 “most” of the time. The contour ‖1+𝜇(𝑡)𝜆(𝑡)‖=1 is termed the Hilger Circle. Since the solution of 𝑧^Δ(𝑡)=𝜆(𝑡)𝑧(𝑡) is 𝑥(𝑡)=𝑧₀𝑒_𝜆(𝑡,𝑡₀), Theorem 4.2 states that if 𝜆(𝑡)∈𝒮(𝕋) then some 𝐾(𝑡₀)>1 exists such that ‖𝑧(𝑡)‖=‖𝑧₀‖‖𝑒_𝜆(𝑡,𝑡₀)‖≤‖𝑧₀‖𝐾𝑒^{−𝛼(𝑡−𝑡0)}, (4.4) where 𝑒_𝜆 is a generalized time scale exponential. The Hilger Circle will be important in the upcoming Lyapunov analysis, as will the following lemmas.

Lemma 4.3. Let 𝑧∈𝐶_rd(𝕋) be a function which is known to satisfy the inequality 𝑧^Δ(𝑡)≤𝜆(𝑡)𝑧(𝑡), 𝑧(0)=𝑧₀∈ℝ with 𝜆∶𝕋→ℝ and 𝜆∈𝐶_rd(𝕋). If 𝑧(𝑡)>0 for all 𝑡≥𝑡₀∈𝕋, then 𝜆(𝑡)∈ℛ⁺.

Proof. Defining 𝑣(𝑡)=𝑧^Δ(𝑡)−𝜆(𝑡)𝑧(𝑡) gives rise to the initial value problem 𝑧^Δ(𝑡)−𝜆(𝑡)𝑧(𝑡)=𝑣(𝑡),𝑧(0)=𝑧₀,𝑡∈𝕋, (4.5) where 𝑣(𝑡)≤0.
First, suppose 𝜆(𝑡) is negatively regressive for 𝑡≥𝑡₀, that is, 𝜆(𝑡)<−1/𝜇(𝑡) with 𝜇(𝑡)>0. Then (4.5) yields 𝑧^𝜎=𝜇𝜆𝑧+𝑧+𝜇𝑣=𝑧(1+𝜇𝜆)+𝜇𝑣<0.
On the other hand, suppose 𝜆(𝑇) is nonregressive for some 𝑇>𝑡₀,𝑇∈𝕋. If 𝜆(𝑡)<−1/𝜇(𝑡) over 𝑇>𝑡≥𝑡₀, then invoke the preceding argument. If 𝜆(𝑡)>−1/𝜇(𝑡) over 𝑇>𝑡≥𝑡₀, then solve (4.5) to get 𝑧(𝑡)=𝑒_𝜆(𝑡,𝑡₀)𝑧₀+^𝑡_𝑡0𝑒_𝜆(𝑡,𝜎(𝜏))𝑣(𝜏)Δ𝜏,𝑇>𝑡≥𝑡₀. (4.6) Since 𝑒_𝜆(𝑡,𝑡₀)>0 for 𝑡<𝑇, we see that ∫^𝑡_𝑡0𝑒_𝜆(𝑡,𝜎(𝜏))𝑣(𝜏)Δ𝜏<0. However, for 𝑡≥𝑇, (4.6) becomes 𝑧(𝑡)=0+_𝑡^𝜌(𝑇)₀𝑒_𝜆(𝑡,𝜎(𝜏))𝑣(𝜏)Δ𝜏,𝑡≥𝑇. (4.7) Thus, 𝑧(𝑡)<0 for all 𝑡≥𝑡₀ for both negatively regressive and nonregressive 𝜆(𝑡), a contradiction of the Lemma's presupposition. This leaves only 𝜆(𝑡)∈ℛ⁺.

At this point we pause briefly to discuss Lyapunov theory on time scales. DaCunha produced two pivotal works [14, 15] on solutions 𝑃 of the generalized time scale Lyapunov equation,

𝐴(𝑡)^𝑇𝑃(𝑡)+𝑃(𝑡)𝐴(𝑡)+𝜇(𝑡)𝐴^𝑇(𝑡)𝑃(𝑡)𝐴(𝑡)=−𝑄(𝑡),𝑡∈𝕋, (4.8) where 𝐴(𝑡),𝑃(𝑡),𝑄(𝑡)∈ℝ^𝑛×𝑛, 𝐴 and 𝑄 are known and 𝑄(𝑡)>0. Though it will not be necessary to solve (4.8) in this work, we will see that the form of (4.8) leads to an upper bound on the graininess that is generally applicable to MIMO systems, an advancement beyond previous works which gave an explicit bound only for SISO systems. DaCunha proves that a positive definite solution 𝑃(𝑡) to the time scale Lyapunov equation with positive definite 𝑄(𝑡) exists if the eigenvalues of 𝐴(𝑡) are in the Hilger circle for all 𝑡∈𝕋. Furthermore, 𝑃(𝑡) is unique. As with the well-known result from continuous system theory (c.f. [16]), the solution is constructive, with

𝑃(𝑡)=_𝕊𝑡Φ^𝑇_𝐴(𝑡)(𝑠,0)𝑄(𝑡)Φ_𝐴(𝑡)(𝑠,0)Δ𝑠, (4.9) where Φ_𝐴(𝑡,𝑡₀) denotes the transition matrix for the linear system 𝑥^Δ(𝑡)=𝐴(𝑡)𝑥(𝑡), 𝑥(𝑡₀)=𝐼. The correct interpretation of this integral is crucial: for each 𝑡∈𝕋, the time scale over which the integration is performed is 𝕊_𝑡∶=𝜇(𝑡)ℕ₀, which has constant graininess for each fixed 𝑡. (Since this paper was written, we have explored the subject on algebraic and dynamic Lyapunov equations in more detail; we refer the interested reader to [17].)

Before the next lemma, we define

𝒦∶={𝑘∶𝕋→ℝ⁺∣𝑘∈𝐶_rd(𝕋),𝑘^Δ(𝑡)≥0∀𝑡∈𝕋,lim_𝑡→∞𝑘(𝑡)=∞}. (4.10) The next lemma follows directly.

Lemma 4.4. Given assumptions (A1) and (A2) and 𝑘∈𝒦, there exists a nonzero graininess 𝜇(𝑡) and a time 𝑡^∗ such that, for all 𝜇(𝑡)≤𝜇(𝑡) and 𝑡>𝑡^∗, the matrix −𝑘𝐶𝐵 satisfies a time scale Lyapunov equation with 𝑃=𝐼, 𝑄(𝑡)≥𝜀₂ for small 𝜀₂>0, and 𝐵 from (3.2).

Proof. We construct 𝜇(𝑡) as 1𝜇(𝑡)∶=(𝜆𝑘(𝑡)^min{𝐶𝐵+(𝐶𝐵)^𝑇}𝜆_max{(𝐶𝐵)^𝑇𝐶𝐵}−𝜀₁),𝑡∈𝕋, (4.11) with 𝜀₁>0 sufficiently small so that 𝜇(𝑡)>0 on 𝕋. This holds if (𝐶𝐵)^𝑇+𝐶𝐵−𝜇(𝑡)𝑘(𝑡)(𝐶𝐵)^𝑇𝐶𝐵=𝜀₁𝐼. (4.12) Multiplying (4.12) by −𝑘(𝑡) gives (−𝑘(𝑡)𝐶𝐵)^𝑇+(−𝑘(𝑡)𝐶𝐵)+𝜇(𝑡)(−𝑘(𝑡)𝐶𝐵)^𝑇(−𝑘(𝑡)𝐶𝐵)=−𝑘(𝑡)𝜀₁𝐼. (4.13) (We now drop the explicit time-dependence for readability.) Set Σ(𝜇)∶=expc(𝜇𝐴)−𝐼, (4.14) so that 𝐵(𝜇)=[𝐼+Σ(𝜇)]𝐵, yielding (−𝑘𝐶𝐵)^𝑇+(−𝑘𝐶𝐵)+𝜇(−𝑘𝐶𝐵)^𝑇(−𝑘𝐶𝐵)=−𝑘(𝜀₁𝐼−𝑍), (4.15) where each term of 𝑍 is a product of contants times Σ(𝜇). Since ‖Σ(𝜇)‖→0 as 𝜇→0 (c.f. Appendix), there exists a time 𝑡^∗∈𝕋 when ‖𝑍‖<𝜀₁. Because the preceding arguments admit any graininess 𝜇(𝑡)≤𝜇(𝑡), it follows that (−𝑘𝐶𝐵)^𝑇+(−𝑘𝐶𝐵)+𝜇(−𝑘𝐶𝐵)^𝑇(−𝑘𝐶𝐵)=−𝑘𝑄,𝑡>𝑡^∗,𝜇(𝑡)≤𝜇(𝑡), (4.16) where 𝑄(𝑡)≥𝜀₂ and 𝜀₂∶=𝜀₁−‖𝑍‖>0.

We comment briefly on the intuitive implication of Lemma 4.4. Equation (4.16) shows that there exist positive definite 𝑄(𝑡) and 𝑃=𝐼 to satisfy an equation of the form (4.8). According to DaCunha, this implies that the eigenvalues of −𝑘𝐶𝐵 lie strictly within the Hilger circle for 𝑡>𝑡^∗.

5. System Stability

We now come to the three central theorems of the paper. If 𝐵𝐶 is not known to be full rank, or cannot be full rank because of the input/output dimensions, then it must be assumed (or determined a priori) that the eigenvalues of 𝐴−𝑘(𝑡)𝐵𝐶 attain negative real parts at some point in time. This phenomenon is investigated in-depth by other authors [1, 4]. We then make use of the observation by Owens et al. [3] that there must exist some 𝑘^∗>0 such that if 𝑘(𝑡)≡𝑘^∗ the system of (A1) has a positive-real realization. This, together with the Kalman-Yakubovich Lemma [16], implies existence of 𝑃,𝑄>0 so that

(𝐴−𝑘^∗𝐵𝐶)^𝑇𝑃+𝑃(𝐴−𝑘^∗𝐵𝐶)=−𝑄,𝑃𝐵=𝐶^𝑇𝐶𝐵. (5.1)

Theorem 5.1 (Exponential Stability). In addition to (A1) and (A2), suppose (i)𝑡∈𝕋 where 𝕋 is a time scale which is unbounded above but with 𝜇(𝑡)≤𝜇(𝑡), (ii)𝑘∈𝒦 (implying from (4.11) that 𝜇(𝑡)→0, but not necessarily monotonically), (iii)𝐵𝐶 is not necessarily full rank, but there exists a time 𝑡^∗∈𝕋 such that the eigenvalues of (𝐴−𝑘(𝑡)𝐵𝐶) are strictly in the left-hand complex plane for 𝑡≥𝑡^∗. Then the system (3.1), is exponentially stable in the sense that there exists time 𝑡₀∈𝕋 and constants 𝐾(𝑡₀)≥1, 𝛼≥0, such that ‖𝑥(𝑡)‖≤‖𝑥(𝑡₀)‖𝐾𝑒^{−(1/2)𝛼(𝑡−𝑡0)},𝑡≥𝑡₀. (5.2)

Proof. Set 𝑘^∗∶=𝑘(𝑡^∗). Then assumption (iii) is the prerequisite for (5.1). Again, we suppress the time-dependence of 𝑥, 𝑘, 𝜇, 𝐴, and 𝐵. Similarly to Lemma 4.4, terms containing Σ(𝜇) may be added to the first equality in (5.1) to obtain (𝐴−𝑘^∗𝐵𝐶)^𝑇𝑃+𝑃(𝐴−𝑘^∗𝐵𝐶)≤−𝜀₃,𝑡≥𝑡^∗∗, (5.3) for some small 𝜀₃>0. Note 𝑡=𝑡^∗∗ is the point at which terms involving Σ(𝜇) become small enough for (5.3) to hold. Defining 𝑍₁∶=𝐶^𝑇𝐶Σ𝐵−𝑃Σ𝐵, the second equality in (5.1) gives 𝑃𝐵=𝐶^𝑇𝐶𝐵+𝑍₁. (5.4)
Consider the Lyapunov function 𝑉=𝑥(𝑡)^𝑇𝑃𝑥(𝑡) with 𝑃 from (5.1). Then, using (5.1), (5.3) and Lemma 4.4, 𝑉^Δ=𝑥^𝑇Δ𝑃𝑥+𝑥^𝜎𝑇𝑃𝑥^Δ=𝑥^𝑇[𝒜^𝑇𝑃+𝑃𝒜+𝜇𝒜^𝑇𝑃𝒜]𝑥=𝑥^𝑇[(𝐴−𝑘^∗𝐵𝐶)^𝑇𝑃+𝑃(𝐴−𝑘^∗𝐵𝐶)−(𝑘−𝑘^∗)(𝐶^𝑇𝐵^𝑇𝑃+𝑃𝐵𝐶)+𝜇(𝐴−𝑘𝐵𝐶)^𝑇𝑃(𝐴−𝑘𝐵𝐶)]𝑥≤𝑥^𝑇[−𝜀₃−(𝑘−𝑘^∗)(𝐶^𝑇𝐵^𝑇𝐶^𝑇𝐶+𝐶^𝑇𝐶𝐵𝐶+2𝑍₁𝐴𝐶)+𝜇^𝑇𝑃𝐴−2𝜇𝑘𝐴^𝑇𝑃𝐵𝐶+𝜇𝑘²𝐶^𝑇𝐵^𝑇𝐶^𝑇𝐶𝐵𝐶+𝜇𝑘²𝐶^𝑇𝐵^𝑇𝑍₁𝐶]𝑥=𝑥^𝑇[−𝜀₃+𝜇𝐴^𝑇𝑃𝐴]𝑥+𝑦^𝑇[−𝑘(𝐶𝐵)^𝑇−𝑘𝐶𝐵+𝜇𝑘²(𝐶𝐵)^𝑇𝐶𝐵+𝜇𝑘²𝐵^𝑇𝑍₁+𝑘^∗((𝐶𝐵)^𝑇+𝐶𝐵)]𝑦+𝑥^𝑇[−(𝑘−𝑘^∗)2𝑍₁−2𝜇𝑘𝐴^𝑇𝑃𝐵]𝑦≤−(𝜀₃−𝜇‖𝐴^𝑇𝑃𝐴‖)𝑥^𝑇𝑥−(𝑘𝜀₂−𝜇𝑘‖𝐵‖‖𝑘𝑍₁‖−2𝑘^∗‖𝐶𝐵‖)𝑦^𝑇𝑦+(2‖𝑘𝑍₁‖+2𝜇𝑘‖𝐴^𝑇𝑃𝐵‖)𝑥^𝑇𝑦. (5.5) At this point, we observe the following. (a)‖𝐴^𝑇𝑃𝐴‖ is bounded because 𝐴 is bounded. Set 𝛾₁∶=sup_𝑡∈𝕋‖𝐴^𝑇𝑃𝐴‖.(b)𝜇𝑘 is bounded by assumption (i) and (4.11).(c)‖𝑘𝑍₁‖ is bounded because 𝑘 is proportional to 1/𝜇 and (1/𝜇)‖Σ(𝜇)‖→𝑐𝑜𝑛𝑠𝑡 as 𝜇→0. Set 𝛾₂∶=sup_𝑡∈𝕋2‖𝑘𝑍₁‖.(d)Set 𝛾₃∶=sup_𝑡∈𝕋𝜇𝑘‖𝐵‖𝛾₂.(e)Set 𝛾₄∶=sup_𝑡∈𝕋2𝑘^∗‖𝐶𝐵‖.(f)Set 𝛾₅∶=sup_𝑡∈𝕋2𝜇𝑘‖𝐴^𝑇𝑃𝐵‖.
Recalling the standard inequality 𝑥^𝑇𝑦≤𝛽𝑥^𝑇𝑥+(1/𝛽)𝑦^𝑇𝑦 for any 𝛽>0, we then have 𝑉^Δ≤−(𝜀₃−𝜇𝛾₁−𝛽𝛾₂−𝛽𝛾₅)𝑥^𝑇𝑥−(𝑘𝜀₂−𝛾₃−𝛾₄−1𝛽𝛾₂−1𝛽𝛾₅)𝑦^𝑇𝑦. (5.6) By assumption (ii), there exists a time 𝑡₀≥max{𝑡^∗,𝑡^∗∗} such that, for sufficiently small 𝛽, 𝑉^Δ≤−(𝜀₃−𝜇𝛾₁−𝛽𝛾₂−𝛽𝛾₅)𝛾_min(𝑃)𝑉∶=𝜂(𝑡)𝑉,𝑡≥𝑡₀, (5.7) with 𝜂(𝑡)<0. Then, by [11, Theorem 6.1], Theorem 4.2, and Lemma 4.3 it follows that there exists 𝐾(𝑡₀)≥1 so that ‖𝑥(𝑡)‖≤𝐾‖𝑥(𝑡₀)‖𝑒^{−(1/2)𝛼(𝑡−𝑡0)},𝑡≥𝑡₀, (5.8) where 𝛼 is defined in (4.2).

We point out that, when 𝐵𝐶 is full rank, assumption (iii) above is no longer necessary as there always exists a 𝑘^∗ such that the eigenvalues of (𝐴−𝑘(𝑡)𝐵𝐶) are strictly real-negative for 𝑘(𝑡)≥𝑘^∗. One more lemma is required before the next theorem.

Lemma 5.2 .. If 𝕋 is a time scale with bounded graininess (i.e., 𝜇_∞∶=sup_𝑡∈𝕋𝜇(𝑡)<∞), then 𝑐₁^∞_𝑡0𝑒^𝛼𝑡𝑑𝑡≤^∞_𝑡0𝑒^𝛼𝑡Δ𝑡≤𝑐₂^∞_𝑡0𝑒^𝛼𝑡𝑑𝑡, (5.9) where 𝑐₁, 𝑐₂, 𝛼∈ℝ and 𝑐₁, 𝑐₂>0.

Proof. Consider the case when 𝛼>0. The process of time scale integration is akin to the approximation of a continuous integral via a left-endpoint sum of (variable width) rectangles. If the function to be summed is increasing (as in this case), the sum of rectangular areas will be less than the continuous integral, meaning 𝑐₂=1, 𝑐₁<1. One estimate of the lower bound, then, follows by simply increasing 𝑐₁ until 𝑐₁𝑒^𝛼𝑡 meets one of the rectangle right endpoints which are given by 𝑒^{𝛼𝜌(𝑡)}. Thus 𝑐₁𝑒^𝛼𝑡≤𝑒^{𝛼𝜌(𝑡)}, or equivalently, 𝑐₁𝑒^{𝛼𝜎(𝑡)}≤𝑒^𝛼𝑡. This in turn yields 𝑐₁≤𝑒^{−𝛼𝜇(𝑡)}. Therefore, the most conservative bound is given by 𝑐₁≤𝑒^{−𝛼𝜇∞}. The case for 𝛼<0 can be argued similarly, leading to the lemma's following conclusion: 𝑐₁=𝑒^{−𝛼𝜇∞}𝑐,𝛼>0,1,𝛼≤0,²=𝑒1,𝛼≥0,_{−𝛼𝜇∞},𝛼<0. (5.10)

We are now in a position to state the main theorem of the paper.

Theorem 5.3. In addition to (A1) and (A2), assume the prototypical update law, 𝑘^Δ(𝑡)=‖𝑦(𝑡)‖² with 𝑘₀∶=𝑘(0)>0. Then lim_𝑡→∞𝑘(𝑡)<∞ and 𝑦∈𝐿₂[𝑡₀,∞)_𝕋.

Proof. For the sake of contradiction, assume 𝑘(𝑡)→∞ as 𝑡→∞. Then 𝑘∈𝒦. Theorem 5.1 yields 𝑥∈𝐿_∞[𝑡₀,∞)_𝕋 and therefore 𝑦∈𝐿_∞[𝑡₀,∞)_𝕋. The solution for 𝑘(𝑡) is (by [11, Theorem 2.77]), 𝑘(𝑡)=𝑘(𝑡₀)+^𝑡_𝑡0‖𝑦(𝜏)‖²Δ𝜏≤𝑘(𝑡₀)+‖𝐶‖^2𝑡_𝑡0‖𝑥(𝜏)‖²Δ𝜏,𝑡≥𝑡_0. (5.11) In conjunction with Lemma 5.2, this allows lim_𝑡→∞𝑘(𝑡)≤𝑘(𝑡₀)+‖𝐶‖^2∞_𝑡0‖𝑥(𝑡)‖²Δ𝑡≤𝑘(𝑡₀)+‖𝐶‖²‖𝑥(𝑡₀)‖²𝐾^2∞_𝑡0𝑒^{−𝛼(𝑡−𝑡0)}Δ𝑡=𝑘₀+‖𝐶‖²‖𝑥(𝑡₀)‖²𝐾^2∞₀𝑒^−𝛼𝑡Δ𝑡<∞. (5.12) This contradicts the assumption, so it must be that 𝑘(𝑡)<∞ for 𝑡∈𝕋. It also immediately follows that ∫^∞_𝑡0‖𝑦‖²Δ𝑡<∞.

It seems possible that Theorem 5.3 may be improved to show that the output is convergent, that is, 𝑦(𝑡)→0 as 𝑡→∞. This is left as an open problem.

6. Discussion

We remark here that there is a great amount of freedom in the choice of the update law for 𝑘(𝑡) (c.f. [5]). We use the simplest choice for convenience (as do most of authors); the essential arguments remain unchanged for other choices. There is also freedom in the choice of update for 𝜇(𝑡). The expression (4.12) can be simplified to

(𝐶𝐵)^𝑇+𝐶𝐵−𝜇(𝑡)𝑘(𝑡)(𝐶𝐵)^𝑇𝐶𝐵>0,𝑡∈𝕋, (6.1) for any 𝜇(𝑡)≤𝜇(𝑡). In the SISO case, this further reduces to the expression derived by Owens [5], that 𝜇𝑘𝐶𝐵<2. It requires the graininess (which may interpreted as the system sampling step size for sample-and-hold systems) to share at least an inverse relationship with the gain, but is otherwise quite unrestrictive. Ilchmann and Townley [7] note that 𝜇(𝑡)=1/(𝑘(𝑡)log𝑘(𝑡)) meets 𝜇𝑘𝐶𝐵<2 after sufficient time without knowledge of 𝐶𝐵. While previous works have always constructed a monotonically decreasing step size, the time scale-based arguments in this paper reveal even greater freedom: 𝜇(𝑡) may actually increase, jump between continuous, and discrete intervals, or even exhibit bounded randomness. Two examples of the usefulness of this freedom come next.

For the first example, we point out that the notation in the previous sections somewhat belies the fact that the system's time domain (its time scale) may be fully or partially discrete, and thus there is no guarantee in Theorem 5.3 that the output has stabilized between samples. As pointed out in [7] and elsewhere, a sampled system with period 𝜇 is detectable if and only if

𝜆_𝑘−𝜆_𝑙2𝜋𝑗𝜇∉ℤforany𝜆_𝑘≠𝜆_𝑙,𝑗=√−1, (6.2) where 𝜆_𝑘,𝜆_𝑙∈{0,spec(𝐴)}. We next comment on how to circumvent the intrasample stabilization problem. Both of the following methods essentially permit the graininess 𝜇 to “wiggle” a bit so that an output sample must eventually occur away from a zero-crossing. Recalling that 𝜇(𝑡)≤𝜇(𝑡) from Theorem 5.1(i), let 𝜇(𝑡)=𝜇(𝑡)𝑣(𝑡), where 0<𝑣≤1is one of the sequences below:

(1) Let 𝑣 be consist of an infinitely repeated subsequence with 𝑛!+1 elements that are random numbers between 0 and 1. Let these elements, labeled 𝑣₁,𝑣₂,…, be irrational multiples of each other. Assume 𝜇(𝑡) converges to 𝜇𝑣₁>0 such that 𝑦(𝑡)=0 but the true continuous output is nonzero. This implies that there exist integers 𝑘,𝑙 such that (𝜆_𝑘−𝜆_𝑙)𝜇𝑣₁/2𝜋𝑗∈ℤ. As sequence 𝑣 advances, there may be at worst 𝑛! combinations of 𝑗,𝑘 such that (𝜆_𝑗−𝜆_𝑘)𝜇𝑣_𝑟/2𝜋𝑗∈ℤ for 𝑟=1⋯𝑛!. However, at the next instant in time, (𝜆_𝑘−𝜆_𝑙)𝜇𝑣_𝑛!+1/2𝜋𝑗 must be irrational and therefore not in ℤ. The controller will detect a nonzero output and continue to adapt 𝑘(𝑡) and 𝜇(𝑡). In practice, of course, it is not possible to obtain a sequence of truly irrational numbers, but most modern computer controllers have enough accuracy to represent the ratio of two very large integers, so that this technique would only fail for impractically high magnitudes of ‖𝜆_𝑘−𝜆_𝑙‖.

(2) Let 𝑣 be a sequence of random numbers in the specified range. Even in a computer with only 8-bit resolution for 𝑣, the probability of (𝜆₁−𝜆₂)𝜇𝑣/2𝜋𝑖∈ℤ drops drastically after a few sample periods.

We remark here that if (6.2) holds, then (𝒜,𝐶) is detectable because (𝐴,𝐶) is detectable. Thus, the stability of 𝑦 implies the stability of 𝑥. We do not dwell on this here, but see, for example, [7] for a similar argument.

For the second example, we consider a problem posed by distributed control networks (c.f. [18, 19]). Here, one communication network supports many control loops as well as a certain volume of unrelated high-priority traffic. The (unpredictable) high-priority traffic may block the control traffic at times, forcing longer-than-anticipated sample periods. In normal operation, the controller may sample fast enough to behave, for all practical purposes, like a continuous control with 𝜇(𝑡)=0. At the blocking instant, 𝑡_𝑏, 𝜇(𝑡_𝑏) rises to some unpredictable level. The scheduling question is when should the blocked controller emit a communications packet of high enough priority to override the block? One answer is straightforward: just before 𝜇(𝑡_𝑏) exceeds 𝜇(𝑡_𝑏), the known maximum sample period that will guarantee plant stability. (In [19] we suggest that even longer delays are possible under certain conditions.) Intuitively, lower gains permit longer blocking delays.

The example of the previous paragraph is closely modeled using the variable ℙ[𝑎,𝑏] time-scale, which is continuous for an interval 𝑎, then has a gap for an interval 𝑏, then repeats. Figure 1 shows the regulation of a system implementing an adaptive gain controller in a blocking situation:

(a)

(b)

Figure 1 

In graph (A), output 𝑦(𝑡) (thick line) and gain 𝑘(𝑡) are plotted. In graph (B), 𝜇(𝑡) (thick line) is kept at 90% of 𝜇(𝑡) during blocking. A block occurs after every 1 second of continuous control.

11𝐴=01−11,𝐵=,𝐶=[1,0]. (6.3) The gain begins at 𝑘₀=1/2 and the sampling period at 𝜇(0)=0. The bounding function for the graininess is 𝜇(𝑡)=1.9/𝑘(𝑡) (so that 𝜇𝑘𝐶𝐵<2).

7. Conclusions

In summary, the paper illustrates a new unified continuous/discrete approach to the high-gain adaptive controller. Using developments in the new field of time-scale theory, the unified results reveal that this type of feedback control works well on a much wider variety of time scales than explored in previous literature, including those that switch between continuous (or nearly continuous) and discrete domains or those without monotonically decreasing graininess. A simulation of an adaptive controller on a mixed continuous/discrete time scale is also given. It is our hope that time-scale theory may find wider application in the broad fields of signals and systems as it seems that many of the tools needed in those fields are beginning to appear in their generalized forms.

Appendix

We comment on the properties of the “expc” function referenced in the main body of the paper.

Lemma A.1. The power series (3.3) has the following properties: (1)expc(𝑋)𝑋=𝑋expc(𝑋).(2)expc(𝑋)=(𝑒^𝑋−𝐼)𝑋⁻¹ when 𝑋⁻¹ exists.(3)For real, scalar arguments 𝑥, expc(𝑥)=𝑒^𝑖𝑥/2sinc(𝑥/2), where sinc denotes the sine cardinal function. (This is the motivation for the expc notation.)(4)‖expc(𝑋)‖≤exp(‖𝑋‖).(5)‖expc(𝑋)−𝐼‖≤exp(‖𝑋‖)−1.

Proof. Parts 1–3 follow immediately from the definition. To verify 4, note ‖^∞_𝑛=1𝑋^𝑛−1𝑛!‖≤^∞_𝑛=1‖𝑋‖^𝑛−1𝑛!≤^∞_𝑛=1‖𝑋‖^𝑛−1(𝑛−1)!=exp(‖𝑋‖). (A.1) Part 5 follows from a similar argument. Note that, by property 5, the decomposition expc(𝜇𝑋)=𝐼+Σ(𝜇) gives ‖Σ(𝜇)‖→0 as 𝜇→0, with uniform convergence.

Acknowledgments

This work was supported by NSF Grants no. EHS-0410685 and CMMI no. 726996 as well as a Baylor University Research Council grant. The authors thank their colleague, Robert J. Marks II, for his very helpful suggestions throughout this project.