Multi-Mode Load Control
3.4 Mode Transition Control
3.4.4 Discrete Switching with the Bumpless Transfer
Dwell-time switch
Compared to the above continuous switching, discrete switching can be still utilized because of the compatibility for different hardware layouts and less energy consump- tion, but an alternative must be conducted to eliminate the transient instability due to fast transfer. To slower the fast transfer, each stabilizing subsystem in the loop is kept for a windowing period, such that the transient effects can dissipate for a sufficiently long time. In Fig. 3.15a, the pressure fluctuation during switch instants causes the instability of discrete switching. In comparison, a windowing period τ is added to implement the mode transfer in Fig. 3.15b. Then, there is enough time to decay the transient instability of pressure, and thereby prevent the infinitely fast switch. The windowing period τ is defined as the dwell time, which transfers the discrete switching to a dwell-time switch as:
Fl (t)≡ Fl[(k − 1)·τ] t ∈ [(k − 1) · τ, k · τ ] (3.12) where k ∈ [1, 2, 3, …].
The principle of the dwell-time switch seems simple, but a new problem refers to how to dwell time τ:
(1) If the value of τ is too small, the infinitely fast switch could not be relieved, and the mode switching is still unstable;
(2) If the value of τ is too large, the mode is switched with a serious delay, and the system response decreases.
Therefore, neither a smaller nor larger value of τ will be disabled to improve the dynamics performance of mode transition. To verify it, switching with a large dwell time τ is shown in Fig. 3.16. After approximately 5 s, the overrunning load decreases to a low level, which can not drive the actuator to move. The system switches to Nor.
mode after a large dwell time, rather than immediate switching. With such a long waiting time to switch, the actuator velocity continues to decrease, which cannot meet the requirements of control performance.
Fig. 3.15 Effect of the dwell-time switch
Fig. 3.16 Actuator velocity with a large dwell time τ
Dynamic optimization of dwell time
Dynamic dwell time design theorem
A dwell-time design approach is presented to capture an optimal value for arbitrary switching instant dynamically. First, the state-space equation of the switched systems with N subsystems is defined as:
˙
x(t) = Aσ(t)x(t) (3.13) where σ:[0, ∞] →I = {1, 2, …, N} is the switching signal, state x ∈ Rn; the matrices Ai ∈ Rn×n; i ∈ I, {t0, t1, t2, … tk, …} denotes the switching sequence where t0 is the initial time and tk denotes the kth switching instantly. The solution of the system (3.2) is assumed to exist and to be unique.
Definition 3.1 Assuming each subsystem is stable, there exists a positive definite symmetric matrix Pi that solves the Lyapunov equation for each σth subsystem.
Second, based on the switching signals σ(t), the positive definite symmetric matrices Pi are patched together to build a global Lyapunov function:
Vσ (ti )(x(ti ))= xT Pσ (ti )x (3.14) AT σ (ti )Pσ (ti )+ Pσ (ti )Aσ (ti )= −Q (3.15) To analyze the stability with constrained switching, a very intuitive multiple Lyapunov functions (MLF) result is usually pursued [20–22]. As illustrated in Fig. 3.17, the switched system is asymptotically stable if the values of the Lyapunov function at the mode-switching instants form a decreasing sequence.
Third, the dynamic dwell time for the switched system (3.2) is then designed in terms of the above MLF stability theorem.
3.4 Mode Transition Control 37
Fig. 3.17 MLF result for an asymptotically stable switched system
Definition 3.2 T(tk) is called the dwell time of the switching signal σ for the kth switching instant, which stands for the time between the switching instants tk and tk+1, i.e., τ(tk) = tk+1 − tk.
Accordingly, the following result is derived by assuming that all the subsystems are stable.
Theorem 3.1 Considering switched system (3.2), if there is a collection of continuous positive scalar functions Vi(x) and numbers λi > 0 satisfying:
α1(||x||) ≤ Vi (x) ≤ α2(||x||) (3.16)
Vi (x) ≤ −λi ·V˙i (x) (3.17) where α1, α2 are class K functions [23], then assume that the system switches from subsystem i to subsystem j at the switching instant tk, k= 1, 2, …., If the dwell time satisfies:
τ(tk) >
(I n(u
i )
λ j if Vj (x(tk)) > Vi (x(tk))
0 if Vj (x(tk))≤ Vi (x(tk)) ui = Vj (x(tk))
Vi (x(tk)) (3.18) Then the switched system (3.2) is asymptotically stable.
In this theorem, the sequences V(tk−), k = 1, 2, …, are proved to be strictly decreasing [23]. In terms of Lyapunov’s second method, the Lyapunov function value for each subsystem itself decreases because all the subsystems are stable. As a conse- quence, the variations of Lyapunov function values are shown in Fig. 3.18, which infers that switched system (3.2) is stable in terms of the MLF stability theorem in Fig. 3.17.
Dynamic dwell time design
Theorem 3.1 provides an effective way to dynamically design the dwell time.
However, the switched system in the IMCS must be linear and homogeneous, as formularized in the system (3.2). Owing to the typical nonlinear characteristic in the hydraulic system, it is necessary to linearize each subsystem in the independent
Fig. 3.18 The Lyapunov functions values at the switching instants in Theorem 3.1
metering control at first. Taking Nor. mode, for instance, the linearization of the electro-hydraulic control system is depicted in Fig. 3.19. When the meter-in valve is regulated by the flow mapping and pressure difference feedback, the flow-pressure gain of this valve is close to zero, such that a constant flow in the inlet could be considered [15]. Besides, the pressure of the rod chamber is controlled to a low value to save energy (approximately 2 bar), and thereby the meter-out valve is omitted to assume that the rod chamber is directly connected to the drain line [15]. At last, differential equations of the Nor. mode is derived as:
mtv˙c = Aapa − Bpvc − Fl (3.19) Va
βe
˙
pa = Qa − Aavc (3.20)
The state-space equation is derived as:
˙
x(t)= A1x(t)+ B1u(t) (3.21)
Fig. 3.19 The linearization process of Nor. mode
3.4 Mode Transition Control 39
x(t)= [vc
pa
]
˙ x(t)=
[v˙c
˙ pa
] , u(t)=
[Qa
Fl ]
(3.22)
A1 =
[−mBp t m Aa t
−AVaβa e 0 ]
B1 = [
0 − m1 t
βe
Va 0 ]
(3.23)
Flo. and the Reg. modes both utilize differential hydraulic circuits, in which two chambers of the actuator are connected to the supply or drain line simultaneously.
Thus, the asymmetric actuator, such as a cylinder, can be considered a discrete transformer with two possible states of operation. Taking Flo. mode, for instance, the linearization of the system is depicted in Fig. 3.20. A plunger cylinder is employed to simplify the actual asymmetric one, and its inlet is also regarded as a constant flow. Referring to Nor. mode, the linearized state-space equations of Flo. mode is derived by:
u(t)= [Qa
Fl
]
(3.24)
A6 =
[ −mBp t mAd t
−AVdβd e 0 ]
B6 =
[ 0 m1
(1−κ)βe t
Vd 0 ]
(3.25)
where the effective pressurized area and the compressible fluid volume in the control volume will change according to the:
Ad =Aa − Ab, Vd = Va + Vb, Qd =Qa − Qb =(1 −κ)Qa (3.26) A similar approach is taken to derive the state-space equation of Reg. mode:
A2 =
[ −mBp t mAd t
−AVdβd e 0 ]
B2 = [
0 −m1 t
(1−κ)βe
Vd 0
]
(3.27)
Fig. 3.20 The linearization process of Flo. mode
Remark The following unified formula is proposed for linearized expressions of a multi-mode IMCS:
˙
x(t)=Aσ (t)x(t)+ Bσ (t)u(t) (3.28) After linearization, input vectors u(t)/= 0 in Eq. (3.28) become another obstacle to designing the dwell time by employing Theorem 3.1. The control input u(t) is utilized to achieve system stability or certain performances. Here a state feedback u(t) = Kσ(t)x(t) is considered, where Kp, ∀ σ(t) = i ∈ {1, 2, …, N}, is the control gain to be determined. Then, the resulting closed-loop system is given by:
˙
x(t) = Aσ(t)x(t) (3.29) where Aσ(t)= Aσ(t)+ Bσ(t)Kσ(t).
To solve the control gain Kp, the following theorem is derived.
Theorem 3.2 Consider the switched linear systems (3.2), λi > 0, ui > 0, I = {1, 2, . . . , N}, If there exist matrices Ui > 0, Ti > 0, i ∈ I, j ∈ I, I /= j ,
AiUi + BiTi + Ui AT i + Ti T Bi T +λiUi ≤ 0 (3.30)
Uj ≤ uiUi (3.31)
Then, for any switching signal, there is a set of stabilizing controllers to guarantee a global, uniformly, and exponential for system (3.2). In addition, control gains can be obtained if Eqs. (3.30) and (3.31) have a solution:
Ki = Ti · Ui−1 (3.32)
where Ui = Pi−1 Uj = P−j 1 Ti = Ki · Pi −1
The demonstration of Theorem 3.2 refers to Ref. [24].
Summary: following steps are listed to dynamically design a dwell time for multi-mode switching:
(1) Referring to Eqs. (3.19)–(3.33), formulate the IMCS before and after mode switching as the standard expression:
˙
x(t)= Aσ (t)x(t)+ Bσ (t)u(t) (3.33) (2) Seek the state feedback of controller gain Kp and Pi, Pj until the matrices satisfy
the conditions of Eqs. (3.30) and (3.32).
(3) Calculate λi using Eqs. (3.16)–(3.17).
(4) Compute dwell time τ(tk) for switching instant tk according to Eqs. (3.18) and (3.31).
3.4 Mode Transition Control 41
Table 3.2 The parameters of
the switching instant Parameters Value
mt (kg) 550
Aa (m2) 0.003849
Ab (m2) 0.002592
βe (MPa) 700
Va0 (m3) 0.01
Vb0 (m3) 0.01
Bp (N s/m) 10,000
vc (m/s) 0.05
pa (MPa) 0.4
Qa (L/min) 11.547
Fl (N) −1800
The arm motion in Fig. 3.10 is employed again to verify the presented dwell time design approach. A dwell time of 0.25 s is set based on the parameters of the switching instant in Table 3.2. Figure 3.21 compares the actuator velocities using three switching methods: discrete, long dwell-time, and the presented approach.
The results verify that the frequent mode switch is eliminated, and few velocity oscillations or sharp decreases occur using the presented approach. A minor drop in the velocity can be put down to the crossing of the neutral spool position during mode switching. The comparison also depicts that a stable mode switching, together with an increase in the system response, can be guaranteed by the presented dwell-time design approach.