Closed-Loop Frequency Analysis of Reset Control Systems

This article introduces a closed-loop frequency analysis tool for reset control systems. To begin with sufficient conditions for the existence of the steady-state response for a closed-loop system with a reset element and driven by periodic references are provided. It is then shown that, under specific conditions, such a steady-state response for periodic inputs is periodic with the same period as the input. Furthermore, a framework to obtain the steady-state response and to define a notion of closed-loop frequency response, including high order harmonics, is presented. Finally, pseudosensitivities for reset control systems are defined. These simplify the analysis of this class of systems and allow a direct software implementation of the analysis tool. This methods gives deeper insight into the performance of the system than that achieved with the describing function method.


I. INTRODUCTION
Proportional Integral Derivative (PID) controllers are used in more than 90% of industrial control applications [1], [2]. However, cuttingedge industrial applications have control requirements that cannot be fulfilled by PID controllers. To overcome this problem, linear controllers may be substituted by nonlinear ones. Reset controllers are one such type of nonlinear controllers, which have attracted attention due to their simple structure and their ability to improve closed-loop performance [3]- [18].
A traditional reset controller consists of a linear element the state of which is reset to zero when its input equals zero. The simplest reset element is the Clegg Integrator (CI), which is an integrator with a reset mechanism [3]. To provide design freedom and applicability, reset controllers such as first-order reset elements [19], [20] and second-order reset elements have been introduced [13]. These reset elements are utilized to construct new compensators to achieve significant performance enhancement [16], [21]- [24]. In order to further improve the performance of reset control systems several techniques, such as the considerations of nonzero reset values [8], [20], reset bands [25], [26], fixed reset instants, and P I + CI configurations [27], [28] have been introduced.
Frequency-domain analysis is preferred in industry, since it allows ascertaining closed-loop performance measures in an intuitive way. In addition, frequency-domain analysis gives valuable information on the steady-state behaviour of the system. However, the lack of such methods for nonlinear controllers is one of the reasons why non-linear controllers are not widely popular in industry. The describing function (DF) method is one of the few methods for approximately studying nonlinear controllers in the frequency-domain and this has been widely used also in the literature of reset controllers [8], [16], [24], [29]. The DF method relies on an approximation of the steady-state output of a nonlinear system considering only the first harmonic of the Fourier series expansion of the input and output signals (assumed periodic). The general formulation of the DF method for reset controllers is presented in [29], which however does not provide any information on the closedloop steady-state response.
In this article, first, sufficient conditions for the existence of the steady-state response for a closed-loop system with a reset element and driven by a periodic input are given. Then, a notion of closed-loop frequency response for reset control systems, including high-order harmonics, is introduced. Pseudosensitivities to combine harmonics and facilitate analyzing reset control systems in closed-loop configuration are then defined. All of these ideas are utilized to develop a toolbox, which is briefly discussed. Note finally that, contrary to the DF method, which provides only approximations for the periodic steady-state response of reset control systems, the proposed tools allow computing exact steady-state responses to periodic excitations.
This article is organized as follows. In Section II, sufficient conditions to define a notion of frequency response are presented. Then, a method to obtain closed-loop frequency responses for reset control systems, including high-order harmonics, is developed, and pseudosensitivities are defined. In Section III, the steady-state response of reset controllers to periodic inputs is studied. Finally, Section IV concludes this article.

II. CLOSED-LOOP FREQUENCY RESPONSE OF RESET CONTROL SYSTEMS
Consider the single-input single-output control architecture in the top diagram of Fig. 1. This includes as particular cases all schemes discussed in Section I. The closed-loop system consists of a linear plant with transfer function G(s), two linear controllers with proper transfer function C L 1 (s) and C L 2 (s), and a reset controller with base transfer function C R (s). Let L be the LTI part of the system and assume that G(s) is strictly proper. The state-space realization of L is described by where ζ(t) ∈ R np describes the states of the plant and of the linear controllers (n p is the number of states of the linear part), A, B, C, B u , C e R , C u , D u , and D e R are the corresponding dynamic matrices, y(t) ∈ R is the output of the plant and w(t) = [r(t) d(t)] T ∈ R 2 is an external input. The state-space representation of the reset controller is given by the equations ⎧ ⎪ ⎨ ⎪ ⎩ẋ The closed-loop state-space representation of the overall system can, therefore, be written as with D L 2 the feedthrough matrix of C L 2 (s).

A. Stability and Convergence
In this section, sufficient conditions for the existence of a steady-state solution for the closed-loop reset control system (3) driven by periodic inputs are provided. This is based on the H β condition [4], [30]- [32], which we recall in what follows. Let The H β condition states that the reset control system (3) with w = 0 is quadratically stable if and only if there exist ρ = ρ T > 0 and β such that the transfer function is strictly positive real (SPR), (Ā, B 0 ) and (Ā, C 0 ) are controllable and observable, respectively, and Definition 1: A timeT > 0 is called a reset instant for the reset control system (3) if e R (T ) = 0. For any given initial condition and input w, the resulting set of all reset instants defines the reset sequence {t k }, with t k ≤ t k+1 , for all k ∈ N. The reset instants t k of the reset control system (3) have the well-posedness property if for any initial condition x 0 and any input w, all reset instants are distinct, and there exists a λ > 0 such that for all k ∈ N, λ ≤ t k+1 − t k [8], [33].
Remark 1: If the H β condition holds, then the reset control system (3) has the uniform bounded-input bounded-state (UBIBS) property and the reset instants have the well-posedness property [34]. Therefore, the reset control system (3) has a unique well-defined solution for t ≥ t 0 for any initial condition x 0 and input w which is a Bohl function [8], [33].
To develop a frequency-domain analysis method for the reset control system (3), the following assumption is required.
Assumption 1: The initial condition of the reset controller is zero. In addition, there are infinitely many reset instants and lim The second term in Assumption 1 is introduced to rule out a trivial situation. In fact, if lim k→∞ t k = T K , then for all t ≥ T K the reset control system (3) is a stable linear system. Two important technical lemmas, which are used in the proof of the following theorem, are now formulated and proved.
Lemma 1: Let {t k } and {t k } be the reset sequences of the reset control system (3) for two different initial conditions ζ 0 andζ 0 of the linear part and for the same input. Suppose Assumption 1 and the H β condition hold and w is a Bohl function. Then lim Proof: To begin with note that, for any initial condition with Since the linear part of the system contains the internal model (8) of w I , and w is a Bohl function, based on [4], [31] e R is asymptotically independent of w I . This implies that lim Lemma 2: Consider the reset control system (3). Suppose Assumption 1 holds, w is a Bohl function, and the H β condition is satisfied. Then, the reset control system (3) is uniformly exponentially convergent.
Proof: To begin with note that the property of uniformly exponentially convergence is as given in [35]. Since the H β condition is satisfied, according to Remark 1, the reset control system (3) has a unique well-defined solution for any initial condition x 0 and any w which is a Bohl function. Let x andx be two solutions of the reset control system (3) corresponding to the some input w and to two different initial conditions. Since the H β condition is satisfied x(t) andx(t) are bounded for all t. Let Δx := x −x, and let {t k } and {t k } be the reset sequences of Moreover, by the well-posedness property, there exists a λ > 0 such that λ ≤ t k+1 − t k and λ ≤t k+1 −t k . Thus, selecting δ sufficiently small yields for all t k sufficiently large. By (9), Now, by (10) and (11) and, using (9) The same discussion applies tot k . Hence, for t sufficiently large we have Due to the satisfaction of the H β condition [4], [30], [31], there exist a matrix P ∈ R (np+nr )×(np+nr ) , P = P T > 0, and a scalar α > 0 such that Using the candidate Lyapunov function Thus, using (13) and (15) for t sufficiently large yields Ξ(t, δ) ≤ 0. Hence, since Δx is bounded, there exist α m > 0 and K > 0 such that for all t ≥ 0 (see Lemma 3 in the Appendix). This implies that the reset control system (3) is uniformly exponentially convergent. Theorem 1: Consider the reset control system (3). Suppose Assumption 1 holds, w(t) = w 0 sin(ωt) 1 , and the H β condition is satisfied. Then, the reset control system (3) has a periodic steady-state solution, which can be expressed asx(t) = S(sin(ωt), cos(ωt), ω) for some function S : R 3 → R nr +np .
Proof: Since the H β condition holds and w(t) = w 0 sin(ωt) is a Bohl function, by Remark 1 the reset control system (3) has a unique solution for any initial condition x 0 . In addition, the reset control system (3) has the UBIBS property and, according to Lemma 2, it is uniformly exponentially convergent. Hence, the proof of the existence of the function S relies on the results in [35]. We only need to show that S is unique. To this end, similarly to [36], assume that the reset control system (3) has two steady-state solutionsx 2 (t) = S 2 (sin(ωt), cos(ωt), ω)(t) and Since the H β condition holds, by Lemma 2 there exist α m > 0 and K > 0 such that hence, the claim. Corollary 1: Consider the reset control system (3) with r(t) = r 0 sin(ωt) and d = 0, for all t ≥ 0. Then, the even harmonics and the subharmonics of the steady-state response have zero amplitude, and the sequence of reset instants is periodic with period π ω .
Proof: The response of (3) for r = r 0 sin(ωt) and d = 0, for all t ≥ 0, is given by where ψ(t) = (ωI cos(ωt) +Ā sin(ωt))F Thus According to [37], uniformly convergent systems forget their initial conditions. By Lemmas 1 and 2, ξ s and the reset instants are unique for any t 0 and ζ 0 . Hence, the transient response of ξ s converges to zero, which implies that Therefore, since reset occurs when if {t s , t s−1 , . . ., t s−m } are reset instants and satisfy (24), then {t s , t s−1 , . . ., t s−m } + π ω also satisfy (24), which implies that the sequence of reset instants is periodic with period π ω . Using this property in (21) shows thatx(t) = −x(t + π ω ) and t s+q − t s = π ω , hence ξ s = −ξ s+q . This means that the even harmonics of the steady-state response of the reset control system (3) have zero amplitude. In addition, , which implies that the steady-state response of the reset control system (3) does not contain any subharmonic. Remark 2: The reset sequence {t k } and the reset values ζ k are independent of the input amplitude for r(t) = r 0 sin(ω t ). We now show that the function S is derived explicitly for r(t) = r 0 sin(ω t ) and d = 0. Suppose that there are q − 1 reset instants between t s and t s + π ω (see Fig. 2). Assume sin(ωt s ) = κ, then cos(ωt s ) = ± √ 1 − κ 2 (without loss of generality, we consider the positive value here). Using trigonometry relations, one has that Moreover with i = 1, 2, . . ., q and g 0 (κ, ξ s ) = ξ s . Now, since e R (t) is zero at reset instants, one has that with i = 1, 2, . . ., q. In addition Moreover, by the well-posedness property of reset instants (see Definition 1), reset instants are distinct. Hence, there are q + 2 independent equations and q + 2 parameters (κ, ξ s , q, τ 1 , τ 2 , . . ., τ q ), q ∈ N. In addition, the well-posedness property implies that the reset intervals are lower bounded [8]. Hence Furthermore, for q = 1, the equations have always a unique solution.
Thus, there exists a bounded nonempty set Q = {Q i ∈ N|Q i ≤ q max } such that for q ∈ Q, the equations have a solution. Hence,x, the steadystate response of the reset control system (3) to r(t) = r 0 sin(ωt), is the solution of (27)-(28) for q = q max . Sincex is periodic with period 2π ω , one has a n cos((2n + 1)ωt) + b n sin((2n + 1)ωt).
Finally, one could also use De Moivre's formula to find a formal polynomial expansion for S in terms of sin(ωt) and cos(ωt).

B. HOSIDF of the Closed-Loop Reset Control Systems
In Section II-A, sufficient conditions for the existence of the steadystate solution for the reset control system (3) driven by periodic inputs have been presented. Moreover, the steady-state solution has been explicitly calculated. In this section, the HOSIDF technique [38] is applied to the steady-state response of the system to derive a notion of frequency response for the reset control system (3), which allows analyzing tracking and disturbance rejection performance (see the bottom diagram of Fig. 1).

1) Tracking Performance:
Consider the reset control system (3) with r(t) = r 0 sin(ωt) and d(t) = 0, for all t ≥ 0. We now derive relations between the input r and the steady-state response of the output y, of the error e, and of the control input u. To this end, consider the steady-state reset instants t s , t s+1 , . . ., t s+q and their associated reset values ξ s , ξ s+1 , . . ., ξ s+q , which are calculated through (27) and (28).  n even (32) in which Proof: The proof requires a straightforward calculation, hence, it is omitted (see also [39]). Note that ψ(t) is defined in (20). Definition 3: The families of complex valued functions S n (jω) and CS n (jω), n = 1, 2, . . ., are the sensitivity and the control sensitivity of the reset control system (3), respectively.
2) Disturbance Rejection: In this section, relations between d(t) = sin(ωt) and the error e and the control input u are found in the case in which r(t) = 0 for the reset control system (3) using the same procedure provided in Section II-B1. To this end, the matrix ψ(t) has to be replaced by ψ D (t) = (ωI cos(ωt) +Ā sin(ωt))F D Let t s , t s+1 , . . ., t s+q and ξ s , ξ s+1 , . . ., ξ s+q be the steady-state reset instants and their associated reset values for the reset control system (3) with d(t) = d 0 sin(ωt) and r(t) = 0, respectively. In addition, since r(t) = 0, (27) is changed tō with i = 1, 2, . . ., q . Now, substituting ψ(t) with ψ D (t) in the relations (25) and (26), and considering (38) instead of (27), the steady-state response of the reset control system (3) for d(t) = d 0 sin(ωt) and r(t) = 0 is found using the same procedure provided in Section II-A.
in which Definition 4: The families of complex valued functions P S n (jω) and CS dn (jω), n = 1, 2, . . ., are the process-sensitivity and the control Fig. 3. Error signal e(t) with its 1st, 3rd, and 5th harmonics. e m (t) is fitted to e(t) and it is an indicator of the maximum error of the system. sensitivity due to the presence of the disturbance of the reset control system (3), respectively.

C. Pseudosensitivities for Reset Control Systems
The analysis of the error signal e and of the control input u is one of the main factors while designing a controller. In linear systems this analysis is performed using the closed-loop transfer functions [40]. As discussed in Section I, although reset control systems may be analyzed using the DF of the reset controller in the closed-loop sensitivity equations, this yields an approximation, which is not precise due to the existence of high order harmonics. On the other hand, it is not trivial to analyze reset controllers considering all harmonics. In order to perform the analysis of reset control systems straightforwardly, we combine all harmonics into one frequency function for each closed-loop frequency response. In the literature, there are several definitions of Bode plot for nonlinear systems [41], [42]. However, all of these focus only on the gain of the system. In the following, pseudosensitivities, which have both gain and phase components, are defined.
It has been proven that the error and the control input signals of the reset control system (3) are periodic with period 2π ω (see Fig. 3). We define the pseudosensitivity as the ratio of the maximum error of the reset control system (3), for r(t) = r 0 sin(ωt) and d(t) = 0, for all t ≥ 0, to the amplitude of the reference at each frequency.
Using (3) and (21), t ext is obtained froṁ Similarly, the pseudoprocess sensitivity, the pseudocomplementary sensitivity, the pseudocontrol sensitivity, and the pseudocontrol sensitivity of the disturbance can be defined (for more detail see [39]). We conclude this section, with the following statement. Corollary 6: Consider the reset control system (3). The pseudosensitivities and the closed-loop HOSIDFs are independent of the amplitude of the harmonic excitation input.
Remark 3: The presented results have been integrated into an open source toolbox, which has been developed using MATLAB, see [43]. This toolbox facilitates the analysis and design for reset control systems.

III. PERIODIC INPUTS
In Section II, a notion of frequency response and pseudosensitivities for reset control systems have been defined. These serve as graphical tools for performance analysis of reset controllers. The pseudosensitivities determine how a system amplifies harmonic inputs at various frequencies, information which is essential for control designers. However, this information is obtained for a single harmonic excitation and since the superposition principle does not hold, it provides only an approximation in the case of multiharmonics excitation. In this section the steady-state performance in the presence of multiharmonics excitation and periodic inputs is investigated. This is reasonable since most references and disturbances are periodic [11]. For ease of notation let lcm( Theorem 3: Consider the reset control system (3). Suppose the H β condition and Assumption 1 hold. Then for any periodic excitation of the form the reset control system (3) has a periodic steadystate solution of the form Proof: Let t s M +i be the steady-state reset instants of the reset control system (3) for w is given in (43). By (3), the steady-state solution for w as in (43) is given bȳ By Lemma 2, the reset control system (3) forgets the initial condition; thus, using a procedure similar to the one in Section II-A, yields Since the reseting condition is  (45) holds, which implies that the sequence of reset instants is periodic with period 2π ω M ; hence,x(t) =x(t + 2π ω M ), and using the Fourier series representation yieldsx (t) = a 0 + ∞ n=1 a n cos(nω M t) + b n sin(nω M t). (46) We conclude this section with the following statement the proof of which can be found in [39].
Corollary 7: Consider the reset control system (3). Suppose the H β condition and Assumption 1 hold. Then for any periodic input w P (t) = w P (t + T P ) the reset control system (3) has a steady-state periodic solution with the same period T P 2 .

IV. CONCLUSION
This article has proposed an analytical approach to obtain closedloop frequency responses for reset control systems, including highorder harmonics. To this end, sufficient conditions for the existence of the steady-state solution of the closed-loop reset control systems driven by periodic inputs have been presented. Moreover, pseudosensitivities, which serve as a graphical tool for performance analysis of reset controllers, have been defined: these relate the error and control input of the system to the reference and the disturbance. All calculations are performed in a user-friendly toolbox to make this approach easy of use. The proposed method predicts the closed-loop performance of reset control systems more accurately than the DF method.

APPENDIX A
Lemma 3: Consider a positive and bounded function V (t). Suppose that there exists α > 0 such that If for t sufficiently large Ξ(t, δ) ≤ 0 then there exist α m > 0 and K > 0 such that Proof: Since V is bounded, by (47) and (48), V achieves its maximum value at some time t vm < ∞. In other words, there exists a time 0 ≤ t vm < ∞ such that 2 Due to lack of space, it is not possible to have illustrative examples in this article. In [39], there are several practical example which show the effectiveness of the proposed approach. Therefore, by (48) and well-posedness property, there exists a bounded set T = {t i > t vm | t i / ∈ M ∧ Ξ(t i , δ) > 0, i ∈ N}. Thus, using (50) there exists a bounded set A = {α i > 0| V (t i ) = e −α i (t i −tv m ) V (t vm ), t i ∈ T }. Since the set A is bounded, there exists a α > 0 such that for all α i ∈ A one has that α ≤ α i . Now considering α m = min(α, α ), based on (47) and (48), yields Finally, if T and A are empty sets, then selecting α m = α the claim yields.