Relay Feedback Analysis for Double Integral Plants

Abstract

Double integral plants under relay feedback are studied. Complete results on the uniqueness of solutions, existence, and stability of the limit cycles are established using the point transformation method. Analytical expressions are also given for determining the amplitude and period of a limit cycle from the plant parameters.

1. Introduction

Relay feedback forms one important class of nonlinear systems and can cause complex nonlinear behaviors. Its early analysis can be traced to 1950–1960s, and afterwards, two basic approaches, the time domain approach [1, 2] and the frequency domain approach [3], emerged. Although they are almost identical, the frequency approach is more popular because of its ease of manipulation. However, as a general method for relay analysis, such a frequency approach also has some limits in itself. Firstly, only necessary conditions on the existence and stability of possible limit cycles are obtained. Secondly, the conditions are usually expressed as a summation of infinite items. It is certainly desirable to find both sufficient and necessary conditions of the existence and stability of limit cycles for relay feedback systems, as well as to give such conditions explicitly in terms of system parameters without any requirement on numerical computation.

In our recently published paper [4], we presented a complete relay analysis for a class of servo plants, 𝐺(𝑠)=𝐾/[𝑠(𝑠+𝑎)], (𝑎>0), including the uniqueness of solutions, existence, and stability of limit cycles, and its amplitude and period, using a similar analysis to [5]. Naturally, it is desirable to investigate whether such an analysis can be extended to double integral plants, that is, the case of 𝑎=0. Most people believe that double integral plants have no limit cycle under relay feedback [1, 3], but our analysis shows that this is not true. Actually, a limit cycle for double integral case is observable, which completely depends on the system initial conditions. Moreover, we also find that the analysis for the case of 𝑎>0 in [4] is not applicable to the case of 𝑎=0. This is because for 𝑎>0, the derivative of system output is stable and bounded as 𝐺′(𝑠)=𝐾/(𝑠+𝑎). While for 𝑎=0, 𝐺′(𝑠)=𝐾/𝑠 and the derivative of system output becomes unstable and not bounded anymore. Hence, the extension is not straightforward and the results for double integral plants are quite different from our previous ones.

This paper is organized as follows. Section 2 gives the main result followed by the proofs and a few remarks, where more details and explanations are given. Conclusions are drawn in Section 3.

2. The Result

Consider a double integral plant 𝐺(𝑠)=𝐾𝑠²,(𝐾≠0),(1) whose state-space representation in the controllable canonical form is given by 𝑥̇(𝑡)=0100_{𝐴}𝑥(𝑡)+01_𝐵𝑢(𝑡),(2)𝑦(𝑡)=𝐾0_{𝐶}𝑥(𝑡),(3) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡)]^𝑇∈ℝ², 𝑦(𝑡),𝑢(𝑡)∈ℝ are the state, output, and input of the system, respectively. The plant is under the relay feedback control: 𝑢(𝑡)=⎧⎨⎩𝑢₋if𝑒(𝑡)<𝜀₋,or𝑒(𝑡)≤𝜀₊,𝑢𝑡₋=𝑢₋,𝑢₊if𝑒(𝑡)>𝜀₊,or𝑒(𝑡)≥𝜀₋,𝑢𝑡₋=𝑢₊,(4) where 𝑢₊ and 𝑢₋ are the relay amplitudes, 𝑒(𝑡)=−𝑦(𝑡), 𝜀₊ and 𝜀₋ are the relay hysteresis with 𝜀₋≤𝜀₊. We assume 𝑢₊≠𝑢₋ since otherwise (4) becomes a constant but no longer a relay control. The relay control is depicted in Figure 1. The initial function for 𝑡=𝑡₀ is 𝑢𝑡₀≡⎧⎪⎨⎪⎩𝑢₊if𝑒𝑡₀>𝜀₊,𝑢₋if𝑒𝑡₀<𝜀₋,𝑢₀∈𝒰if𝜀₋≤𝑒𝑡₀≤𝜀₊,(5) where 𝑡₀ is the initial time and 𝒰∶={𝑢₋,𝑢₊}. We call (1)–(5) a relay feedback system (RFS) which is depicted in Figure 2, where 𝐶(𝑠) represents the controller.

Figure 1 

Relay function.

Figure 2 

Relay feedback system.

If the RFS generates a limit cycle, let 𝑇₊ and 𝐴₊ be the half period and the extreme value corresponding to 𝑢(𝑡)=𝑢₊, respectively, and 𝑇₋ and 𝐴₋ be the half period and the extreme value corresponding to 𝑢(𝑡)=𝑢₋, respectively, as shown in Figure 3. 𝑇_𝑟 (resp., 𝑇_𝑑) is the rising (resp., decreasing) period between the switching time, 𝑡_𝑖 (resp., 𝑡_𝑖+1), when the limit cycle trajectory traverses 𝒮₊ (resp., 𝒮₋) and the peak (resp., valley) time.

Figure 3 

Limit cycles.

We are now in a position to state the sufficient and necessary conditions for the existence of solutions, the existence and stability of limit cycles, and the amplitudes and periods of limit cycles.

Theorem 1. Consider the RFS for double integral plant 𝐺(𝑠) in (1). (i)A unique solution exists for any initial condition.(ii)A limit cycle exists if and only if 𝐾𝑢₊>0>𝐾𝑢₋, 𝜀₊=𝜀₋=𝜀 and any of the following holds: (a)𝑥₁(𝑡₀)≠−𝜀/𝐾;(b)𝑥₁(𝑡₀)=−𝜀/𝐾 but 𝑥′(𝑡₀)≠0. If this is the case, the limit cycle is formed after the first switch, which is unique with two switchings per period.(iii)If a limit cycle exists, it is globally marginally stable. (iv)If a limit cycle exists, its amplitude and period are described by 𝐴₊=𝐾𝑥²₂𝑡₀−2𝑢𝑡₀𝐾𝑥₁𝑡₀+𝜀2𝑢₊,𝐴₋=−𝐾𝑥²₂𝑡₀−2𝑢𝑡₀𝐾𝑥₁𝑡₀+𝜀2𝑢₋,𝑇₊=2𝐾²𝑥²₂𝑡₀−2𝐾𝑢𝑡₀𝐾𝑥₁𝑡₀+𝜀𝐾𝑢₊,𝑇₋=−2𝐾²𝑥²₂𝑡₀−2𝐾𝑢𝑡₀𝐾𝑥₁𝑡₀+𝜀𝐾𝑢₋.(6)

Proof. For a RFS, the input 𝑢(𝑡) is a piecewise constant function. Without loss of generality, suppose 𝑢(𝑡)=𝑢₋𝟏𝑡−𝑡₀+^∞_𝑖=1(−1)^𝑖−1𝑢₊−𝑢₋𝟏𝑡−𝑡_𝑖,𝑡∈𝑡₀,∞,𝑡₀<𝑡₁<⋯,(7) as shown in Figure 4, where 𝟏(𝑡)=⎧⎨⎩1,𝑡≥0,0,𝑡<0.(8)Define the switching planes: 𝒮₊∶=𝜉∈ℝ²∶−𝐶𝜉=𝜀₊,𝒮₋∶=𝜉∈ℝ²∶−𝐶𝜉=𝜀₋.(9) If the trajectory of 𝑥(𝑡) traverses 𝒮₊ (resp., 𝒮₋), that is, −𝐶𝑥(𝑡_𝑖)=𝜀₊ (resp., −𝐶𝑥(𝑡_𝑖)=𝜀₋) at some instant 𝑡=𝑡_𝑖>𝑡₀ with −𝐶𝑥(𝑡⁻_𝑖)<𝜀₊ (resp., −𝐶𝑥(𝑡⁻_𝑖)>𝜀₋) and −𝐶𝑥(𝑡⁺_𝑖)>𝜀₊ (resp., −𝐶𝑥(𝑡⁺_𝑖)<𝜀₋), then the instant 𝑡=𝑡_𝑖 is called a switching time. In particular, 𝑡_𝑖 denotes the switching time when the 𝑖th switching takes place.
The state response of (2) to 𝑢(𝑡) in (7) is given by 𝑥(𝑡)=e^{𝐴(𝑡−𝑡0)}𝑥𝑡₀+^𝑡_𝑡0e^{𝐴(𝑡−𝜏)}𝐵𝑢(𝜏)d𝜏=1𝑡−𝑡₀01𝑥𝑡₀+⎡⎢⎢⎣𝑡−𝑡₀²2𝑡−𝑡₀⎤⎥⎥⎦𝑢₋𝟏𝑡−𝑡₀+^∞_𝑖=1(−1)^𝑖−1⎡⎢⎢⎣𝑡−𝑡_𝑖²2𝑡−𝑡_𝑖⎤⎥⎥⎦𝑢₊−𝑢₋𝟏𝑡−𝑡_𝑖,(10) where 𝑡∈[𝑡₀,∞), 𝑡₀<𝑡₁<⋯. It is easy to show that both 𝑥₁(𝑡) and 𝑥₂(𝑡) are continuous at switching time 𝑡_𝑖,𝑖=1,2,…, as 𝑡−𝑡_𝑖=0 for 𝑡=𝑡_𝑖. However, 𝑥̇₂(𝑡⁻₁)=lim_𝑡→𝑡⁻₁𝑥̇₂(𝑡)=𝑢₋, while 𝑥̇₂(𝑡⁺₁)=lim_𝑡→𝑡⁺₁𝑥̇₂(𝑡)=𝑥̇₂(𝑡₁)=𝑢₋+(𝑢₊−𝑢₋)=𝑢₊ indicating 𝑥̇₂(𝑡) is not continuous at 𝑡=𝑡₁ due to 𝑢₊≠𝑢₋ by our assumption. Straightforwardly, this discontinuity persists for the general case of 𝑡=𝑡_𝑖.
If considering the time between two consecutive switchings only, that is, 𝑡∈[𝑡_𝑖,𝑡_𝑖+1), (10) is simplified as 𝑥₁(𝑡)=𝑥₁𝑡_𝑖+𝑡−𝑡_𝑖𝑥₂𝑡_𝑖+𝑡−𝑡_𝑖²2𝜇,𝑥₂(𝑡)=𝑥₂𝑡_𝑖+𝑡−𝑡_𝑖𝜇,(11) where 𝑖=0,1,2,…, and 𝜇=⎧⎨⎩𝑢₊if𝑖isodd,𝑢₋if𝑖iseven.(12) Since 𝑒(𝑡)=−𝑦(𝑡)=−𝐾𝑥₁(𝑡) and 𝑒^(𝑡)=−𝑦^(𝑡)=−𝐾𝑥₂(𝑡), it yields 𝑒(𝑡)=𝑒𝑡_𝑖+𝑡−𝑡₁𝑒^𝑡_𝑖−𝑡−𝑡_𝑖²2𝐾𝜇,(13)𝑒^(𝑡)=𝑒^𝑡_𝑖−𝑡−𝑡_𝑖𝐾𝜇.(14)
It can be seen from (13) that 𝑒(𝑡_𝑖), 𝑒′(𝑡_𝑖) and 𝐾𝜇 will affect behavior of 𝑒(𝑡) and thus determine whether or not 𝑒(𝑡) will reach the switching level of 𝜀₊ or 𝜀₋. Without loss of generality, suppose the initial condition 𝑒(𝑡₀)≤𝜀₊, thus 𝑢(𝑡₀)=𝑢₋. To see how 𝑒(𝑡) evolves with time, the following four cases are considered which are mutually exclusive and cover all possible cases.
Case 1. 𝑒^(𝑡₀)≤0 and 𝐾𝑢₋≥0. For 𝑡∈[𝑡₀,𝑡₁), 𝑒^(𝑡)=𝑒^(𝑡₀)−(𝑡−𝑡₀)𝐾𝑢₋≤0, which implies 𝑒(𝑡)≤𝑒(𝑡₀)≤𝜀₊ so that 𝑥(𝑡) never traverses 𝒮₊ for all 𝑡>𝑡₀. The trajectory of 𝑥(𝑡) is governed by (11) with 𝑡_𝑖=𝑡₀ and 𝜇=𝑢₋.Case 2. 𝑒^(𝑡₀)≤0 and 𝐾𝑢₋<0. For 𝑡∈[𝑡₀,𝑡₁), it follows from (14) that 𝑒^(𝑡)=−𝐾𝑢₋>0, which implies there always exists some 𝑡₁>𝑡₀ such that 𝑒(𝑡₁)=𝜀₊ and 𝑒^(𝑡₁)>0, that is, 𝑥(𝑡) traverses 𝒮₊ at 𝑡=𝑡₁. For 𝑡∈[𝑡₁,𝑡₂), it follows from (14) that 𝑒^(𝑡)=𝑒^(𝑡₁)−(𝑡−𝑡_𝑖)𝐾𝑢₊>0 if 𝐾𝑢₊≤0, which implies 𝑒(𝑡)>𝑒(𝑡₁)=𝜀₊≥𝜀₋ so that 𝑥(𝑡) never traverses 𝒮₋ for all 𝑡>𝑡₁. The trajectory of 𝑥(𝑡) is governed by (13) and (14) with 𝑡_𝑖=𝑡₁ and 𝜇=𝑢₊. If 𝐾𝑢₊>0, it follows from (14) that 𝑒^(𝑡)=−𝐾𝑢₊<0, which implies there always exists some 𝑡₂>𝑡₁ with 𝑒(𝑡₂)=𝜀₋ and 𝑒^(𝑡₂)<0 so that 𝑥(𝑡) traverses 𝒮₋ at 𝑡=𝑡₂. Afterwards, with the same analysis as above, it is straightforward to verify that 𝑥(𝑡) will traverse 𝒮₊ and 𝒮₋ alternatively and consecutively.Case 3. 𝑒^(𝑡₀)>0 and 𝐾𝑢₋<0. For 𝑡∈[𝑡₀,𝑡₁), it follows from (14) that 𝑒^(𝑡)=𝑒^(𝑡_𝑖)−(𝑡−𝑡_𝑖)𝐾𝑢₋>0, which implies there always exists 𝑡₁>𝑡₀ such that 𝑒(𝑡₁)=𝜀₊ and 𝑒′(𝑡₁)>0, that is, 𝑥(𝑡) will traverse 𝒮₊ at 𝑡=𝑡₁. From 𝑡=𝑡₁ onwards, following the same analysis in Case 2, if 𝐾𝑢₊≤0, 𝑥(𝑡) never traverses 𝒮₋ for all 𝑡>𝑡₁, the trajectory of 𝑥(𝑡) is governed by (13) and (14) with 𝑡_𝑖=𝑡₁ and 𝜇=𝑢₊. Otherwise, if 𝐾𝑢₊>0, 𝑥(𝑡) will traverse 𝒮₋ and 𝒮₊ alternatively and consecutively.Case 4. 𝑒^(𝑡₀)>0 and 𝐾𝑢₋≥0. For 𝑡∈[𝑡₀,𝑡₁), it follows from (14) that 𝑒^(𝑡)=−𝐾𝑢₋≤0. If 𝑒^(𝑡)<0, there always exists 𝑡_0𝑚 such that 𝑒(𝑡)≤𝑒(𝑡_0𝑚) for all 𝑡≥𝑡₀. If 𝑒(𝑡_0𝑚)≤𝜀₊, then 𝑒(𝑡)≤𝑒(𝑡_𝑚)≤𝜀₊ so that 𝑥(𝑡) never traverses 𝒮₊ for 𝑡≥𝑡₀. The trajectory of 𝑥(𝑡) is governed by (13) and (14) with 𝑡_𝑖=𝑡₀ and 𝜇=𝑢₋. On the contrary, if 𝑒(𝑡_𝑚)>𝜀₊, there always exists 𝑡₁ satisfying 𝑡₀<𝑡₁<𝑡_0𝑚 such that 𝑒(𝑡₁)=𝜀₊ and 𝑒^(𝑡₁)>0, that is, 𝑥(𝑡) traverses 𝒮₊ at 𝑡=𝑡₁. If 𝑒^(𝑡)=0, then 𝑒^(𝑡)=𝑒^(𝑡₀)>0, which also implies 𝑥(𝑡) traverses 𝒮₊ at some 𝑡=𝑡₁. For 𝑡∈[𝑡₁,𝑡₂), it follows from (14) that 𝑒^(𝑡)=𝑒^(𝑡₁)−(𝑡−𝑡_𝑖)𝐾𝑢₊>0 if 𝐾𝑢₊≤0, which implies 𝑒(𝑡)>𝑒(𝑡₁)=𝜀₊≥𝜀₋ so that 𝑥(𝑡) never traverses 𝒮₋ for all 𝑡>𝑡₁. The trajectory of 𝑥(𝑡) is governed by (13) and (14) with 𝑡_𝑖=𝑡₁ and 𝜇=𝑢₊. If 𝐾𝑢₊>0, it follows from (14) that 𝑒^(𝑡)=−𝐾𝑢₊<0, which implies there always exists some 𝑡₂>𝑡₁ such that 𝑒(𝑡₂)=𝜀₋ and 𝑒^(𝑡₂)<0, that is, 𝑥(𝑡) traverses 𝒮₋ at 𝑡=𝑡₂. For 𝑡∈[𝑡₂,𝑡₃), it follows from (14) that 𝑒^(𝑡)=𝑒^(𝑡₂)−𝐾𝑢₋<0, which implies 𝑥(𝑡) never traverses 𝒮₊ again for all 𝑡>𝑡₂. The trajectory of 𝑥(𝑡) is governed by (14) with 𝑡_𝑖=𝑡₂ and 𝜇=𝑢₋.
In view of the above analysis, the solution to (2) always exists. This completes (i).
We can also see from the analysis in (i) that consecutive switchings between 𝒮₊ and 𝒮₋ take place if and only if 𝐾𝑢₊>0>𝐾𝑢₋. Under this condition, let 𝑡_𝑖 be the switching instant of 𝑥(𝑡) on the plane 𝒮₊, that is, 𝑒(𝑡_𝑖)=𝜀₊ and 𝑒(𝑡)<𝜀₊, 𝑢(𝑡)=𝑢₋ for sufficiently close 𝑡<𝑡_𝑖. This implies 𝑒^(𝑡_𝑖)≥0. The case 𝑒^(𝑡_𝑖)=0 can occur only when 𝜀₋=𝜀₊ and then (𝜀₊,0) is a fixed point of 𝑥(𝑡). But a fixed point is unique. Hence, no limit cycle exists in this case.Assume 𝑒^(𝑡_𝑖)>0, then 𝑢(𝑡) switches from 𝑢₋ to 𝑢₊ at 𝑡=𝑡_𝑖. In Case 2, we have shown that 𝑒^(𝑡) decreases and becomes negative until 𝑥(𝑡) traverses 𝒮₋ at 𝑡=𝑡_𝑖+1. Consider the Poincare mapping 𝜒=𝑒^𝑡_𝑖⟶𝜙₊(𝜒)=𝑒^𝑡_𝑖+1.(15) The function 𝜙₊ is defined on the ray [0,+∞) and takes values in the ray (−∞,0]. Let 𝜒=𝑒^(𝑡_𝑖)>0 and 𝜏=𝑡_𝑖+1−𝑡_𝑖. By definition, 𝑒(𝑡_𝑖)=𝜀₊ and 𝑒(𝑡_𝑖+1)=𝜀₋. Since both 𝑒(𝑡) and 𝑒^(𝑡) are continuous at 𝑡=𝑡_𝑖+1, it follows from (13) and (14), respectively, by taking 𝑡=𝑡_𝑖+1 that 𝜀₋=𝜀₊+𝜏𝜒−𝜏²2𝐾𝑢₊,(16)𝜙₊(𝜒)=𝜒−𝜏𝐾𝑢₊.(17) It follows from (17) that 𝐾𝑢₊=𝜒−𝜙₊(𝜒)𝜏.(18) Substituting (18) into (16) yields 𝜙₊(𝜒)=−𝜒+2𝜀₋−𝜀₊𝜏≤−𝜒<0,(19) which implies ||||𝜙₊(𝜒)𝜒||||≥1.(20) If 𝜀₊>𝜀₋, “>” holds in (20) so that the Poincare mapping (15) is not a contraction and no limit cycles exist. If 𝜀₋=𝜀₊=𝜀, “=” holds in (20), which implies a limit cycle is formed after the first switching. As 𝑒^(𝑡_𝑖)>0, 𝑥^(𝑡₁)≠0 must be satisfied so that (a) and (b) can be easily derived with the help of (11). This completes (ii).
Once a limit cycle is formed, if 𝑒^(𝑡_𝑖) is changed due to some disturbance, 𝑥(𝑡) cannot remain at the original limit cycle, but form a new one with different amplitude and period, which means the limit cycle is marginally stable. Since the initial condition 𝑥(𝑡₀) is arbitrarily chosen, the limit cycle is globally marginally stable. This completes (iii).
It has been shown in (ii) that for 𝐾𝑢₊>0>𝐾𝑢₋, a limit cycle always exists after the first switching. For 𝑡∈[𝑡₁,𝑡₂), it follows from (14) and with the help of (20) that 𝑒^𝑡₂=𝑒^𝑡₁−𝑇₊𝐾𝑢₊=−𝑒^𝑡₁.(21) Thus, 𝑇₊=2𝑒^𝑡₁𝐾𝑢₊,𝑒^𝑡₁>0.(22) Let 𝑒^(𝑡)=𝑒^(𝑡₁)−(𝑡−𝑡₁)𝐾𝑢₊=0, then 𝑡−𝑡₁=𝑒^(𝑡₁)/(𝐾𝑢₊). Substituting it into (13) yields 𝐴₊=𝑒(𝑡)−𝜀=𝑒^𝑡₁²2𝐾𝑢₊.(23) In the similar way, we have 𝑇₋=−2𝑒^𝑡₁𝐾𝑢₋,(24)𝐴₋=−𝑒^𝑡₁²2𝐾𝑢₋.(25) For 𝑡∈[𝑡₀,𝑡₁), it follows from (13) that 𝑒𝑡₁=𝑒𝑡₀+𝑡−𝑡₀𝑒^𝑡₀−𝑡−𝑡₀²2𝐾𝑢𝑡₀=𝜀,(26) which gives 𝑡−𝑡₀=𝑒^𝑡₀−𝑒^𝑡₀²−2𝐾𝑢𝑡₀𝜀−𝑒𝑡₀𝐾𝑢𝑡₀,(27) where negative root is ignored since 𝑒^(𝑡₁)>0. Substituting (27) into (14) yields 𝑒^𝑡₁=𝑒^𝑡₀²−2𝐾𝑢𝑡₀𝜀−𝑒𝑡₀=𝐾²𝑥²₂𝑡₀−2𝐾𝑢𝑡₀𝐾𝑥₁𝑡₀+𝜀.(28) Substituting (28) into (22)–(24), respectively, yields 𝐴₊=𝐾𝑥²₂𝑡₀−2𝑢𝑡₀𝐾𝑥₁𝑡₀+𝜀2𝑢₊,𝐴₋=−𝐾𝑥²₂𝑡₀−2𝑢𝑡₀𝐾𝑥₁𝑡₀+𝜀2𝑢₋,𝑇₊=2𝐾²𝑥²₂𝑡₀−2𝐾𝑢𝑡₀𝐾𝑥₁𝑡₀+𝜀𝐾𝑢₊,𝑇₋=−2𝐾²𝑥²₂𝑡₀−2𝐾𝑢𝑡₀𝐾𝑥₁𝑡₀+𝜀𝐾𝑢₋,(29) which completes (iv).

Figure 4 

A piecewise constant input.

Remark 1. In our previous relay analysis for a class of servo plant as 𝐺(𝑠)=𝐾/(𝑠(𝑠+𝑎)), (𝑎>0), we proved the existence of a limit cycle by showing the Poincare map 𝜙₊ is a strict contraction, that is, |𝜙^₊(𝜒)|<1 for all 𝜒∈(0,+∞). Here for the case of 𝑎=0, differentiating both sides of (16) yields 𝜏^(𝜒)=𝜏𝜏𝐾𝑢₊−𝜒.(30) Differentiating both sides of (16) and using the help of (20) yield ||𝜙^₊(𝜒)||=||1−𝜏^(𝜒)𝐾𝑢₊||=||||𝜒𝜒−𝜏𝐾𝑢₊||||=||||𝜒𝜙₊(𝜒)||||≤1.(31) If 𝜀₊≠𝜀₋, |𝜙^₊(𝜒)|<1 for all 𝜒∈(0,+∞), but no limit cycle exists in this case as we show. It seems to be contradiction there and the reason is that for 𝑎=0, 𝜙₊(𝜒) is not bounded any more. This also shows the previous results for 𝑎>0 cannot be simply extended to the double integral case here by just letting 𝑎→0.

Remark 2. The limit cycle forms after the first switching but is marginally stable. This means the trajectory will not remain at the original limit cycle even if a very small disturbance is introduced. Nevertheless, it will form a new limit cycle with different amplitude and period immediately after the next switching.

Remark 3. The limit cycle is a parabolic curve with the symmetry of 𝑥^(𝑡_𝑖)=−𝑥^(𝑡_𝑖+1), where 𝑡_𝑖 is the switching time for 𝑖=1,2,…. Its amplitude and period are only decided by the initial conditions, which is also consistent with the result from [6].

Example 1. Consider a double integral plant 𝐺(𝑠)=1/𝑠² under the relay feedback with 𝜀=1, 𝑢₊=1.0 and 𝑢₋=−0.8. The initial conditions are assumed to be 𝑥₁(𝑡₀)=0.5 and 𝑥₂(𝑡₀)=1. The simulated response of the system is shown in Figure 5. By Theorem 1(iv), the amplitude and period of the limit cycle occurred are also calculated. The comparison of the results from the theory and the simulation as shown in Table 1, where one can see the error, is quite minor.

Table 1 

Limit cycle characteristics of Example 1.

Figure 5 

A piecewise constant input.

3. Conclusion

In this paper, a double integral plants under relay feedback is addressed. Complete results have been established on uniqueness of solutions, existence, and stability of limit cycles and its amplitude and period. Comparing with the plant 𝐺(𝑠)=𝐾/[𝑠(𝑠+𝑎)], 𝑎>0 in [4], analytical expressions for the amplitude and period of limit cycles with terms of system parameters are available, but the conditions on the existence of limit cycles are quite different. Reasons for such differences as well as some explanations are provided.