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Keywords:

  • periodic adaptation;
  • time-varying parameters;
  • adaptive control;
  • discrete-time system;
  • nonlinear process

SUMMARY

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. DISCRETE-TIME PERIODIC ADAPTIVE CONTROL
  5. EXTENSION TO MORE GENERAL CASES
  6. ILLUSTRATIVE EXAMPLE
  7. CONCLUSION
  8. REFERENCES

A periodic adaptive control approach is proposed for a class of nonlinear discrete-time systems with time-varying parametric uncertainties which are almost periodic, and the only prior knowledge is the periodicity. The new adaptive controller updates the parameters and the control signal periodically in a pointwise manner over one entire period, in the sequel that achieves a bounded tracking convergence. The result is further extended to scenarios with unknown input gain, higher order dynamics, and tracking. Copyright © 2012 John Wiley & Sons, Ltd.

INTRODUCTION

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. DISCRETE-TIME PERIODIC ADAPTIVE CONTROL
  5. EXTENSION TO MORE GENERAL CASES
  6. ILLUSTRATIVE EXAMPLE
  7. CONCLUSION
  8. REFERENCES

Periodic adaptive control is a relatively new idea in the area of adaptive control theory. In periodic adaptive control, the parameters are periodic in nature and can vary either with respect to time or with respect to system states. In the classical adaptive control, the parametric adaptation mechanism essentially consists of a number of integrators, thus the adaptive control system is able to achieve asymptotic tracking convergence in the presence of constant parametric uncertainties, [1-3]. On the other hand, for dealing with a class of time-varying periodic unknown parameters, a method is introduced that is based on pointwise integration relying on the a priori knowledge of the periodicity of the parameters, [4-7].

Periodic variations are encountered in many real systems. These variations can exist in the system parameters, [8, 9], or as a disturbance to the system, [10-12]. This necessitates the effort in formulating an adaptive control scheme that can handle a class of systems with time-varying periodic unknown parameters or disturbances by taking into account the periodicity of the variations.

In certain cases, the variations of parameters may not be exactly periodic and there can exist a bounded drift in every cycle. There can also be cases where there may exist unmodeled bounded uncertainties in the system. In these cases, robustness of the periodic adaptive control may be an issue. In [4] and [5], the parameters are assumed to be perfectly periodic and no discussion was made to ensure robustness in the case of parameter drift or unmodeled uncertainties.

Robustness in adaptive control has been the subject of much research in both continuous-time and discrete-time, because modeling uncertainties may result in poor performance and even instability of the closed-loop system as observed in [13]. To enhance the robustness of the adaptive control system, many update law modifications were proposed, [14, 15]. These methods make the adaptive closed-loop system robust in the presence of an external disturbance or model uncertainties.

In this work, we apply the concept of periodic adaptation with weighing coefficients to discrete-time systems with almost periodic time-varying parameters that have a bounded drift. In particular, we will show that the new periodic adaptive controller can work effectively to nullify the influence from the periodic component of the time-varying parametric uncertainties to the control error and retain robustness in the presence of bounded drift. The boundedness of the parametric estimate and convergence analysis are conducted by using Lyapunov–Krasovsky like functionals in terms of the estimation errors.

The paper is organized as follows. In Section 2, we present the new periodic adaptive control approach and give complete analysis. To clearly demonstrate the underlying idea and method, we consided the simplest nonlinear dynamics with a single time-varying parameter. In Section 3, we extend the new approach to more general cases. The first extension considers multiple time-varying parameters and time-varying gain of the system input. The second extension considers a general tracking control problem. The third extension considers a higher-order system in canonical form. In Section 4, an illustrative example is provided.

Throughout this paper, ∥ · ∥ denotes the Euclidean norm. For notational convenience, in mathematical expressions, fk represents f(k).

DISCRETE-TIME PERIODIC ADAPTIVE CONTROL

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. DISCRETE-TIME PERIODIC ADAPTIVE CONTROL
  5. EXTENSION TO MORE GENERAL CASES
  6. ILLUSTRATIVE EXAMPLE
  7. CONCLUSION
  8. REFERENCES

In order to clearly demonstrate the idea, first, consider a scalar discrete-time system with only one unknown parameter

  • display math(1)

where θk is almost periodic, that is, θk = ϕk + δk where the component ϕk is periodic with a known period N > 1 and the uncertainty | δk | < ϵ for some unknown constant ϵ. The function ξk = ξ(xk) is assumed to be a known nonlinear function that satisfies | ξ(xk) | ⩽ c1 + c2 | xk + 1 | . For simplicity, only the regulation problem will be considered in this section and the extension to tracking tasks will be considered in the next section.

Discrete-time periodic adaptive control revisited

Consider the system (1), if an approach similar to [5] is used, then it is possible to formulate the control and adaptation laws of the form

  • display math(2)
  • display math(3)

where inline image is the estimate of the periodic component ϕk and γ is a positive adaptation gain. Defining inline image and substituting the adaptive law (2) into the dynamical relation (1), the closed-loop system can be expressed as

  • display math(4)

Now, following a similar approach to [5], consider a non-negative function inline image, its difference over one step is

  • display math(5)

From (5), it is not possible to guarantee the negativity of ΔVk, and thus, the boundedness of inline image because of the existence of the term inline image. To solve this problem, a modification to the adaptation law (3) will be introduced that will guarantee the boundedness of inline image and the tracking, or regulation, error convergence to a minimum bound dictated by the size of the uncertainty δk. This is presented in the next section.

Periodic adaptation for almost periodic parameters

Consider the system (1) and revise the adaptive control mechanism (2), (3) into the following periodic one

  • display math(6)
  • display math(7)

where αk is a positive weighing coefficient and inline image is the initial value for the parameter estimate. The time interval is measured such that k = k0 + nN with n > 1 being the total number of periods in the interval [0,k).

Consider the uncertainty δk and assume that ϵ = λη for some tunable constant λ and an unknown η. It is easy to see that | δkξk | ⩽ λη | ξk | and because η is assumed to be unknown, then αk can be defined as [15],

  • display math(8)

where inline image is the estimate of η and λ can be chosen as any constant as long as it satisfies 0 < λ < λmax, with λmax being shown later to satisfy inline image and inline image is defined as inline image. The estimation law for η can be given as

  • display math(9)

Defining inline image, then it is possible to obtain

  • display math(10)

Note that from (8) when inline image, it is possible to obtain

  • display math(11)

Convergence analysis

Consider the parameter estimation errors, defined earlier, inline image and inline image. Substituting the adaptive control (6) into the dynamical relation (1) and subtracting the adaptive law (6) from ϕk = ϕk − N, the closed-loop system, for any k ⩾ N, can be expressed as

  • display math(12)

The convergence property of the periodic adaptive control system (12) is summarized in the following theorem.

Theorem 1. For the closed-loop system (12), the parameter estimation errors inline image and inline image are bounded and the regulation error xk converges to a bound of size inline image, where inline image.

Proof. Similar to the time-invariant case, select a non-negative function inline image, its difference over a single time step for any k ⩾ N is

  • display math(13)

Substitution of (12) and (33) into (14), it is obtained that

  • display math(14)

To proceed further, note that from the delayed form of (12), it is possible to obtain

  • display math(15)

multiplying both sides with xk − N + 1 and by using the fact that inline image, (15) becomes

  • display math(16)

Substitution of (11) and (16) into (14) and using the fact that inline image, it is obtained that

  • display math(17)

Thus, ΔVk is non-increasing, implying that inline image and inline image are bounded and that there exists positive constants inline image and inline image such that inline image and inline image.

Applying (17) repeatedly for any k ∈ [pN,(p + 1)N] and noticing k0 = k − pN, it is obtained that

  • display math(18)

Because k0 ∈ [0,N), and

  • display math

when k[RIGHTWARDS ARROW] ∞ , according to (14)

  • display math(19)

Consider that Vk is non-negative, and inline image is finite in the interval [0,N); thus, according to the convergence theorem of the sum of series, it is obtained that

  • display math(20)

With the use of (20) and the sector condition on ξk, the Key Technical Lemma guarantees that ξk is bounded and consequently αkxk − N + 1[RIGHTWARDS ARROW] 0 as k[RIGHTWARDS ARROW] ∞ . Implying that there must exist a positive constant ρ such that maxj ∈ [0,k]{αjxj − N + 1}⩽ ρ. Then according to the definition of αk in (8),

  • display math(21)

Following the analysis in [15] and the bound on ξk, the maximum tracking error bound is found to be

  • display math(22)

implying that inline image. Because from (20), it is guaranteed that αkxk − N + 1[RIGHTWARDS ARROW] 0 as k[RIGHTWARDS ARROW] ∞ then the error bound as k[RIGHTWARDS ARROW] ∞ is simply

  • display math(23)

and λ can be selected small enough to ensure that the steady state error limk[RIGHTWARDS ARROW] ∞  | xk + 1 | is as small as possible.□

Remark 1. The periodic adaptive controller can be used along with a stabilizing controller to provide much better tracking performance because of the ability of the adaptive controller to handle periodically varying parameters. Thus, with the existence of a stabilizing controller, the finiteness of inline image in the initial phase is obvious.

EXTENSION TO MORE GENERAL CASES

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. DISCRETE-TIME PERIODIC ADAPTIVE CONTROL
  5. EXTENSION TO MORE GENERAL CASES
  6. ILLUSTRATIVE EXAMPLE
  7. CONCLUSION
  8. REFERENCES

In this section, the concept will be extended to multiple periodic parameters, mixed periodic and time-invariant parameters, the trajectory tracking problem, and the higher-order systems, respectively.

Extension to multiple parameters and time-varying input gain

For simplicity, consider a scalar system

  • display math(24)

where inline image are unknown almost periodic parameters, inline image is a known vector valued function. bk ∈ C[0, ∞ ) is a time-varying and uncertain gain of the system input. The prior information with regards to bk is that the control direction is known and invariant, that is, bk is either positive or negative and nonsingular for all k. Without loss of generality, assume that bk ⩾ bmin, where bmin > 0 is a known lower bound. Similar to the previous scalar case, the unknown parameters θ0 and bk are assumed to have a periodic component and a bounded uncertainty such that inline image and bk = βk + ϵk. Because the uncertainties, inline image and ϵk are assumed to be bounded, then the augmented uncertainties inline image is bounded as ∥ δk ∥ ⩽ λη, where λ > 0 is a tunable constant and η is assumed to be unknown.

Note that each unknown parameter, inline image or βk, may have its own period Ni or Nb. The periodic adaptive control will still be applicable if there exists a common period N, such that each Ni and Nb can divide N with an integer quotient. This is always true in discrete-time because Ni and Nb are integers; therefore, N can be either the least common multiple of Ni and Nb, or simply the product of Ni and Nb when both are prime. Therefore, N is used as the updating period. The presence of the uncertain system input gain makes the controller design more complex. To derive the periodic adaptive control law, define inline image to be the estimation of βk and inline image, the system dynamics (24) can be rewritten as

  • display math(25)

By observation, the control law is selected in the form

  • display math(26)

where inline image. Substituting (26) into (25) yields the closed-loop system

  • display math(27)

where inline image, inline image, and inline image. Note that the computation of ξk requires the inverse of the system input gain estimate inline image and may cause a singularity in the solution if the estimate of the system input gain is zero. To ensure that this never occurs, a semi-saturator must be applied on the input gain estimator such that the estimate never goes below the lower bound. For this purpose and based on (27), the adaptation law is

  • display math(28)

where inline image, the matrix Γϕ is a positive definite matrix of dimension m + 1, γ is a positive constant, and αk is a positive weighing coefficient. Define the vector inline image such that

  • display math(29)

then the semi-saturator is defined as

  • display math(30)

Consider the uncertainty δk, similar to the scalar case, it is easy to see that ∥ δkξk ∥ ⩽ λη ∥ ξk ∥ and because η is assumed to be unknown, then αk can be defined as

  • display math(31)

where inline image is the estimate of η and λ can be chosen as any constant as long as it satisfies 0 < λ < λmax, with λmax being defined later. The estimation law for η can be given as

  • display math(32)

Defining inline image, then it is possible to obtain

  • display math(33)

Note that from (8) when inline image, it is possible to obtain

  • display math(34)

The validity of the aforementioned periodic adaption law is verified by the following theorem.

Theorem 2. Under the periodic adaptation law (28), the closed-loop dynamics (27) converges to a bound inline image for some constants d1 and d2.

Proof. The convergence analysis is analogous to the preceding case, selecting a non-negative function inline image. Note that inline image, where a is defined in (29). The difference of Vk with respect to a single time step is

  • display math(35)

Note that, because the actual input gain is assumed to be positive and so should the minimum bound, bmin, the magnitude of the estimation error would be the same or larger if no saturator was implemented because for estimates below bmin, the difference inline image. Thus, it is concluded that

  • display math(36)

,and furthermore, for a positive-definite matrix inline image, then the following is also true

  • display math(37)

Therefore, it is possible to simplify (35) further to the form

  • display math(38)

Using the fact that inline image and (34), it is possible to obtain

  • display math(39)

Following the same steps that lead to (20) in Theorem 1, it is concluded that

  • display math(40)

The result (39) shows that inline image, inline image, and inline image are bounded because Vk is non-increasing and thus the control signal inline image for some constant q. If the nonlinear function is sector-bounded, that is, inline image for some positive constants c1 and c2, then inline image for some positive constants d1 and d2. Thus, establishing the condition required by the key technical lemma that guarantees αkxk − N + 1[RIGHTWARDS ARROW] 0 as k[RIGHTWARDS ARROW] ∞ .

Further, there must exist a positive constant ρ such that maxj ∈ [0,k]{αjxj − N + 1}⩽ ρ. Then, according to the definition of αk in (31),

  • display math(41)

where inline image. Following the analysis in Theorem 1 and the bound on ξk, the maximum tracking error bound is found to be

  • display math(42)

implying that inline image. Because from (20), it is guaranteed that αkxk − N + 1[RIGHTWARDS ARROW] 0 as k[RIGHTWARDS ARROW] ∞ , then the error bound as k[RIGHTWARDS ARROW] ∞ is simply

  • display math(43)

Extension to tracking tasks

Consider the scalar system (24) with multiple unknown parameters and the unknown periodic input gain. It is required that the state, xk, follow a given reference trajectory r(k). Specifying the tracking error as ek = xk − rk, it is obtained that

  • display math(44)

Rewriting (44) in the form

  • display math(45)

To accommodate the tracking task, the periodic adaptive control (26) can be revised as the following.

  • display math(46)
  • display math(47)

where inline image, and inline image. The weighing coefficient αk is also given as

  • display math(48)

and the estimate inline image is given as

  • display math(49)

The closed-loop system for any k ⩾ N is given by

  • display math(50)

Note that the tracking error dynamics in (50) has the same form as (27), and the adaption mechanism (46) also has the same form as (28) with the state xk replaced by the tracking error ek. Thus, Theorem 2 is directly applicable to this case and the asymptotic convergence of the tracking error ek can easily be verified.

Extension to higher-order systems

Finally, consider the cascaded system of the form

  • display math(51)

where xi,k is the i-th system state, i = {1, 2,  ⋯ , n − 1}, xi,k = [x1,k, ⋯ ,xi,k]T, xn,k = [x1,k, ⋯ ,xn,k]T, inline image, bi,k are unknown almost periodic parameters, inline image is a known vector valued function which is sector bounded, inline image (c1 and c2 being arbitrary positive constants). Similar to the previous case, it is assumed that the unknown parameters in every subsystem have periodic components inline image, inline image with a common period Ni and uncertainties inline image, ϵi,k.

Assuming that all the states are available and that Ni > n, the control necessary to achieve x1,k[RIGHTWARDS ARROW] 0 is proposed as

  • display math(52)

where zi,k = [z1,k, ⋯ ,zi,k]T is the state estimate of xi,k and

  • display math(53)

The future zi,k + n − i can be obtained from (53) in the form

  • display math(54)

where Ii × n − 1 = [Ii × i,0] with Ii × i being an identity matrix of size i, inline image and the matrix inline image can be given as

  • display math(55)

The parameter adaptation law is proposed as follows

  • display math(56)

and

  • display math(57)

where inline image, inline image, L[ · ] is the semi-saturator defined by (30), inline image, inline image, and Γi,ϕ, Γn,ϕ are positive definite matrices. The coefficients αi,k and αn,k can be easily found in the same way as in (31). Substitute the control (52) into (51) and performing successive substitutions, it is possible to obtain

  • display math(58)

where inline image, inline image are the state estimation errors and inline image. It is possible to simplify (58) further into the form

  • display math(59)

Note that the state estimation error terms in (59) can be obtained from (51) and (53) in the form

  • display math(60)

where every subsystem is in the form similar to (27). Thus, the derivations and conclusions in Theorem 2 hold and the parameter estimates inline image and inline image are bounded and each state estimation has an error bound as defined in Theorem 2. Clearly, because all the terms on the right-hand side of (59) are bounded, then ultimately, the state x1,k is also bounded.

Remark 2. Extension from first-order to higher-order systems can also be applied to systems with tracking problems as discussed in the preceding subsections.

ILLUSTRATIVE EXAMPLE

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. DISCRETE-TIME PERIODIC ADAPTIVE CONTROL
  5. EXTENSION TO MORE GENERAL CASES
  6. ILLUSTRATIVE EXAMPLE
  7. CONCLUSION
  8. REFERENCES

Consider a system

  • display math(61)

which is a simplified model of a rotorcraft in forward motion, [16], where θk = 0.5 sin(πk / 25) + δk = ϕk + δk and | δk | 0.01. The matrix Γϕ is chosen to be unity, whereas γ = 1 as well. The parameter tuning parameter associated with the uncertainty is λ = 0.01 Let bk = 3 + 0.5 sin(πk / 50) + ϵk = βk + ϵk with | ϵk | 0.01. The minimum common period is N = 100. A periodic adaptive controller is used with the least squares estimator, [5], and compared with the new approach. Figure 1 shows the regulation error over each. As it can be seen, the noise in the parameters causes degradation in the steady state error which is attenuated by using the new approach; however, in the approach proposed in [5], the error diverges at around t = 12s before converging.

image

Figure 1. Regulation error profiles of the system with time-varying parameters using the new approach and the approach in [5].

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Finally, it is required that the rotorcraft velocity vk track a given reference rk which has a steady state value of rk = 10 m / s. Figure 2 shows that the tracking error using the new approach is better than the one proposed in [5].

image

Figure 2. Tracking error profile of the system using new periodic adaptation and the approach in [5].

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CONCLUSION

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. DISCRETE-TIME PERIODIC ADAPTIVE CONTROL
  5. EXTENSION TO MORE GENERAL CASES
  6. ILLUSTRATIVE EXAMPLE
  7. CONCLUSION
  8. REFERENCES

In this paper, a modified adaptive control approach characterized by periodic parameter adaptation is proposed, which complements the existing periodic adaptive control approach presented in [5]. The new approach is found to be robust to parametric uncertainties. Both regulation and tracking problems were discussed. Extension to higher-order processes was also explored. The validity of the proposed approach is confirmed through theoretical analysis and numerical simulations.

REFERENCES

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. DISCRETE-TIME PERIODIC ADAPTIVE CONTROL
  5. EXTENSION TO MORE GENERAL CASES
  6. ILLUSTRATIVE EXAMPLE
  7. CONCLUSION
  8. REFERENCES
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