Abstract
Chapter 4 reviews the concepts of static stability, dynamic stability, eigenanalysis, and Lyapunov analysis required for the development of necessary and sufficient conditions for stability of a class of nonlinear systems, Hamiltonian natural systems. The combination of static stability of conservative systems and dynamic stability of adiabatic irreversible work processes in the form of exergy and entropy rate equations will be utilized to develop Hamiltonian Surface Shaping and Power Flow Control (HSSPFC).
1 Introduction
HSSPFC is a two-step control law design and analysis process. The first step treats the Hamiltonian system as if it were a conservative system with no externally applied nonconservative forces. This process enables the shaping of the Hamiltonian surface with acceleration feedback and/or proportional feedback to create an isolated minimum (stable) energy state. Static stability is utilized to find the first stability boundary, a rigid body mode (singular stiffness matrix), which defines the point of static neutral stability. The second step analyzes and designs controllers for a Hamiltonian system with externally applied nonconservative forces. This step applies dynamic stability concepts to modify the power flow with dissipation and generation feedback. In the previous chapters, the energy balance equation from the First Law of Thermodynamics combined with the entropy balance equation from the Second Law of Thermodynamics are written in rate equation form to create the exergy rate equation. This is equivalent to power flow which is the first time derivative of the Hamiltonian for natural systems (the work/rate equation). The recognition that the Hamiltonian is stored exergy provides the basic relationship necessary to apply the Second Law of Thermodynamics to the power flow to sort terms into three classes: storage, generation, and dissipation. This sorting process identifies the second stability boundary, a limit cycle, which defines the point of dynamic neutral stability.
It will be explained that dynamically stable systems must be statically stable, but the converse is not true. Therefore, static stability is a necessary condition for stability, while dynamic stability is a sufficient condition for stability and produces a maximum entropy state as well as a minimum energy state. The development of this chapter will combine the writings [51] of Meirovitch [6], Chetayev [14], La Salle and Lefschetz [52], Scanlan and Rosenbaum [53], and Robinett and Wilson [7, 13, 46].
2 Static Stability and Dynamic Stability
Basically, the necessary and sufficient conditions for stability of Hamiltonian natural systems, both linear and nonlinear, can be determined from the shape of the Hamiltonian surface and its time derivative, the power flow. The proof of this observation begins with the assistance of the concepts of static stability and dynamic stability.
The concepts of static stability and dynamic stability will be reviewed, built upon, and expanded in this section to provide the basis for a two-step control law design and analysis process referred to as Hamiltonian Surface Shaping and Power Flow Control (HSSPFC). HSSPFC will be formalized in the last section of this chapter.
In aerospace engineering, flight stability of airplanes and missiles is broadly defined by two categories, static stability and dynamic stability, which leads to a two-step design process for implementing the desired handling qualities into an airplane/missile system [31, 54, 55]. To begin the review, the sum of the forces and moments acting on a missile while on a steady, straight flight path must be in static equilibrium. Typically, static stability is considered as a linearization of the system about a prescribed flight path and defined by the following statement:
If the forces and moments on a body caused by a disturbance tend initially to return (move) the body toward (away from) its equilibrium state, the body is statically stable (unstable) [54, 55].
An equilibrium state is an unaccelerated motion wherein the sums of the forces and moments on the body are zero. Static neutral stability occurs when the body is disturbed and the sums of the forces and moments on the body remain zero. This happens when the system has a rigid body mode or zero stiffness in the system. Figure 4.1 graphically presents the concept of static stability. It will become evident in the following description of dynamic stability that a dynamically stable body must always be statically stable, but static stability is not sufficient to ensure dynamic stability [31, 54, 55]. Therefore, static stability is a necessary condition for stability.
A specific example of static stability is found in reentry vehicle flight stability [55]. For an axisymmetric reentry vehicle, the static margin determines the static stability. The static margin (SM) is the difference in length between the center-of-mass and the center-of-pressure relative to the nose (see Fig. 4.2) of the reentry vehicle. For
then the following definitions apply:
If SM<0, the reentry vehicle is statically unstable. If SM>0, the reentry vehicle is statically stable. The aerodynamic moment for a reentry vehicle is presented in Fig. 4.3 (left), which upon integration with respect to the angle of attack, α, gives a quadratic potential function (see Fig. 4.3, right). Clearly, static stability for a reentry vehicle is equivalent to the analysis of the conservative forces applied to a body which is defined by the energy storage surface, the Hamiltonian.
To build and expand upon static stability, the system is treated as though it is a conservative natural Hamiltonian system with no externally applied nonconservative forces or moments, without linearization. The Lagrange–Dirichlet Theorem (a state where the potential energy is an isolated minimum is a stable equilibrium state [52]) can be applied at this point to the energy storage surface, Hamiltonian, which is a constant:
The equations of motion are given in first-order canonical form as
and in second-order form as
A system is statically stable if the Hamiltonian and potential functions are positive definite about the equilibrium state:
for
and
The Converse of the Lagrange–Dirichlet Theorem, Lyapunov’s Theorem (at an isolated maximum of the potential energy, the equilibrium state is unstable [51, 52]), can be applied, and the system is statically unstable if
The Instability Theorem of Chetayev [14] (if at an equilibrium state the potential energy is not a minimum, then the equilibrium state is unstable [52]) can be applied, and the system is statically neutrally stable, a rigid body mode, if
These theorems will be discussed in more detail in Sect. 4.4, which is focused on Lyapunov analysis.
Continuing the review, the sum of the forces and moments acting on a missile while on an unsteady or curved flight path must be in dynamic equilibrium. Typically, dynamic stability is considered as a linearization of the system about a prescribed flight path and defined by the following statement in terms of the time history of the motion of a body after encountering a disturbance:
A body is dynamically stable (unstable) if, out of its own accord, it eventually returns to (deviates from) and remains at (away from) its equilibrium state over a period of time [54, 55].
A dynamically neutral stable body occurs when a limit cycle exists [7, 46]. A dynamically stable body must always be statically stable, but static stability is not sufficient to ensure dynamic stability [31, 54, 55]. Therefore, static stability is a necessary condition for stability, and dynamic stability is a sufficient condition for stability.
To build and expand upon dynamic stability, the system is treated as though it is a natural Hamiltonian system with externally applied nonconservative forces and/or moments that is statically stable without linearization. The equations of motion are given in first-order canonical form as
and second-order form as
The energy and power flow discussions with respect to stability of the previous chapters, in particular Sect. 3.3, are used to determine dynamic stability.
The system path/trajectory traverses a positive definite energy storage surface (statically stable) defined by the Hamiltonian as a result of the power flow. The time derivative of the energy/Hamiltonian surface defines the power flow into, dissipated within, and stored in the system. This determines whether the system is rising to a higher energy state (away from its equilibrium state), dropping to a lower energy state (returning to its equilibrium state), or staying on a closed cyclic path (limit cycle) constrained to the energy/Hamiltonian surface. Average power flow calculations are used because we cannot guarantee that the opposing power flows will cancel or be dominant generators or dissipators point-for-point in time. In fact, limit cycles balance over the cycle (see Chap. 3, Sect. 3.6).
A system is dynamically stable if the power flow on the average drives the perturbed system to a lower energy state which eventually converges to the statically stable equilibrium state,
A system is dynamically unstable if the power flow on the average drives the perturbed system to a higher energy state which eventually diverges from the statically stable equilibrium state,
A system is dynamically neutral stable if the power flow on the average drives the perturbed system to a closed cyclic path (limit cycle) constrained to the energy/Hamiltonian surface which orbits the statically stable equilibrium state,
These concepts will be proved in the next two sections with respect to an extension of eigenanalysis and Lyapunov analysis.
3 Eigenanalysis
The goal of this section is to relate and extend eigenanalysis of linear systems to nonlinear systems via the energy storage surface, power flow, and limit cycles. One place to begin achieving this goal is to discuss “self-excited” systems in the context of aeroelasticity and aircraft flutter. Self-excited systems include feedback controllers because the forces acting on the system are functions of the coordinates and their velocities and accelerations. This discussion follows the presentation in Scanlan and Rosenbaum [53] and utilizes the linearized version of the nonlinear mass, spring, damper system in Fig. 4.4. The Hamiltonian is
with
and equation of motion is
For the linearized system,
the Hamiltonian becomes
and the equation of motion is
The time derivative of the Hamiltonian is
Four feedback controller examples will be discussed next.
First, assume that c=0 and u=−K P x, which is proportional feedback and derivable from a potential function. The Hamiltonian and its derivative become
and
which produces the eigenvalue problem
with the undamped natural frequency (eigenvalue) of a statically stable and dynamically neutral stable system with a second-order center (linear limit cycle) for
On the other hand, if
then the system is statically (and dynamically) unstable and exponentially divergent without oscillation. For k+K P =0, the system is statically neutral stable with a rigid body mode and dynamically unstable.
Second, assume that c=0 and \(u = -K_{A} \ddot{x}\), which is acceleration feedback and derivable from a kinetic energy function. The Hamiltonian and its derivative become
which produces the eigenvalue problem
with the undamped natural frequency (eigenvalue) of a statically stable and dynamically neutral stable system with a second-order center (linear limit cycle) for
On the other hand, if
then the system is statically (and dynamically) unstable and exponentially divergent without oscillation. For m+K A =0, the system is reduced to a kinematic system (see Chap. 7 on Collective Plume Tracing).
Third, assume that \(u=-K_{D} \dot{x}\), which is damping feedback and treated as additional damping from a nonconservative applied force, where
which produces a complex eigenvalue problem with the real part of the eigenvalue defined by the right-hand term in the time derivative of the Hamiltonian where the system is statically stable and dynamically stable if
dynamically unstable if
and dynamically neutral stable if
This damping feedback is either the additional power being dissipated within the system to drive it to the statically stable equilibrium state or additional power flowing into the system that is driving it to a higher energy level diverging from the statically stable equilibrium state. Notice that the damping feedback has no effect on the static stability of the system, which will be addressed further in Sect. 4.4.
Fourth, assume that \(u = -K_{I} \int_{0}^{t} x\, d \tau\), which is integral feedback and treated as a power generator from a nonconservative applied force, where
which produces a complex eigenvalue problem where the system is statically stable and dynamically stable if
dynamically unstable if
and dynamically neutral stable if
The next three subsections will help provide insights on how to determine that the integral control term is a power or exergy generator.
3.1 Integral Feedback Is an Exergy Generator—Comparison to a Lag Stabilized System
In this subsection, the idea of phase shifting a feedback control signal is used to explain the effect of integral feedback in an analogous way to time-delayed feedback. First, compare a PID controller
with Lag-stabilization [57]
Note that from PID the PD portions contribute to system stiffness and damping. What does the “I” or integral portion contribute to? Negative damping. Since lag-stabilization [57] was shown to phase shift x to \(\dot{x}\), some prescribed amount of damping, for the “integration” of x, the phase shift would be proportional to \(-\dot{x}\). These insights will be expanded upon in Chap. 5 when discussing fractional calculus control. Recognizing this, we have
Differentiating yields
for no net damping (neutral stability), and where
we have
3.2 Integral Feedback Is an Exergy Generator—Investigation by Exergy/Entropy Control Stability Boundary
For the mass–spring–damper system with PID control equation (4.39), the stability boundary is at
or
where the averages can be removed for a linear system. Substituting the appropriate terms from (4.39) gives
then rearranging as
and differentiating both sides gives
resulting in
Next, this result is further clarified with a conventional Routh–Hurwitz analysis.
3.3 Integral Feedback Is an Exergy Generator—Routh–Hurwitz Stability Analysis
First, convert the combined mass–spring–damper PID control system into the following equivalent third-order system:
Next, invoking the change of variable \(y = \int_{0}^{t} x \,d \tau\) gives
and transforming to the s-domain yields
For the third-order system to be stable, the Routh–Hurwitz analysis [32] results in the following necessary and sufficient conditions:
-
1.
All the polynomial coefficients must be positive, and
-
2.
$$\biggl( \frac{C+K_D}{M}\biggr) \biggl( \frac{K+K_P}{M}\biggr) >\frac{K_I}{M} .$$(4.54)
The specific equality condition
implies that
This result is the marginal or neutral stability boundary which agrees with the previous lag-stabilized and exergy/entropy control results:
Also note that for PID control numerical simulation results (presented in the next subsection) that Case 1, (4.36) is asymptotically stable, Case 2, (4.38) is neutrally stable, and Case 3, (4.37) is unstable, which matches the analysis results.
3.4 PID Control Design Numerical Example
The PID control law is partitioned into terms of exergy dissipation, exergy generation, and exergy storage. By applying exergy/entropy control design, the time derivative of the Hamiltonian yields for θ=x (rotary mass–spring–damper analogy)
Once again, the first row in (4.56) is identified as a dissipative term composed of the derivative control term along with the damping term. The second row in (4.56) is the integral action and is identified as the generative input. Note that the proportional control term is added to the system stiffness term and contributes to the reversible portion of the third row in (4.56).
For PID control, the stability boundary is determined as
and is also considered a special case of the average power for linear systems. In general for nonlinear systems, the average power terms will need to be taken into account.
Numerical simulations are performed for four separate PID control regulator cases with the numerical values listed in Table 4.1. The rotary mass–spring–damper system is subject to an initial condition of θ 0=1.0 rad and \(\dot{\theta}_{0} = 0.0\) rad/s. For Case 1A, pure generative input is demonstrated where growing state oscillations (see Fig. 4.5) and positively increasing values are observed in the responses for exergy and exergy rate (see Fig. 4.6). The system has experienced pure generative input without any dissipation mechanisms to damp out the exergy coming into the system. For Case 1, the integral of position, position, velocity, and acceleration responses along with the exergy and exergy rate responses are plotted in Figs. 4.7 and 4.8, respectively. For this case, the dissipative term is greater than the generative term. This is observed from the decaying system responses. In Case 2, the system responses along with the exergy and exergy rate responses are shown in Figs. 4.9 and 4.10, respectively. In this case, the dissipative term is equal to the generative term. This results in system responses that do not decay or have constant oscillatory behavior. Further investigation of the exergy and exergy rate responses show that the mirror images for dissipation and generation will cancel each other out. For Case 2, the potential and kinetic energy rate responses (see Fig. 4.11) are shown to have zero contribution or are reversible over each cycle. In Case 3, the system responses along with the exergy and exergy rate responses are shown in Figs. 4.12 and 4.13, respectively. In this case, the dissipative term is less than the generative term. This results in system responses with increasing oscillatory behavior. In addition, the sum of generative and dissipative terms is increasingly positive, hence exergy is being pumped into the system at a greater rate than can be dissipated. In conclusion, Fig. 4.14 shows the responses for the total exergy, and Fig. 4.15 for exergy rates with respect to each case. Again, for Case 2, a balance or boundary (neutral stability) can be observed.
The three cases (1–3) for the PID control regulator rotary mass–spring–damper dynamic system demonstrate the three subcases in Sect. 3.6.1 and [13]: Given a priori \(T_{0} \dot{\mathcal{S}}_{i} > 0\) and \(\dot{W} >0\), the inertial-spring-damper with PID control showed the following:
-
Case 1 yielded \((T_{0} \dot{\mathcal{S}}_{i})_{\mathrm{ave}} > (\dot {W})_{\mathrm{ave}}\), asymptotic stability, damped stable response, and demonstration of (4.36).
-
Case 2 yielded \((T_{0} \dot{\mathcal{S}}_{i})_{\mathrm{ave}} = (\dot {W})_{\mathrm{ave}}\), neutral stability, and demonstration of (4.38). This case is the dividing line where derivative and integral action cancel each other out.
-
Case 3 yielded \((T_{0} \dot{\mathcal{S}}_{i})_{\mathrm{ave}} < (\dot {W})_{\mathrm{ave}}\), increasing unstable system, and demonstration of (4.37).
3.5 The Power Flow Principle of Stability for Nonlinear Systems
It is now appropriate to extend these linear eigenanalysis results to nonlinear systems by following the concepts in [46]. The following nonlinear extension to the previous eigenvalue problems can be discussed with respect to (4.14) and (4.17):
with u=0. The first term describes the nonlinear frequency content [32] of the undamped/undriven system
which effectively extends the undamped natural frequency, (4.24) to nonlinear systems and defines the Hamiltonian surface and the static stability of the system with \(g(x) = \partial\mathcal{V}/ \partial x\) and \(\mathcal{V}(x) > 0\) when x≠0 and \(\mathcal{V}(x) = 0\) when x=0.
The second term describes the dynamic stability of the nonlinear system and the existence of a limit cycle (equations (4.12)–(4.14))
which is an extension of the real part of the eigenvalue, such as equations (4.36)–(4.38), to nonlinear systems. For example, (4.72) shows a nonlinear frequency content (amplitude dependent) and a resulting coupling into the effective damping which produces initial-condition-dependent limit cycle behavior (refer to Figs. 3.22 and 3.24).
An example of a nonlinear Hamiltonian is a cubic nonlinear spring with the following Hamiltonian:
where \(u = -K_{A} \ddot{x} - K_{P_{\mathrm{NL}}}x^{3} + \hat{u}\) and \(m, k, k_{\mathrm{NL}},K_{A}, K_{P_{\mathrm{NL}}} > 0\). This system is statically stable due to the positive definite Hamiltonian surface with the associated nonlinear proportional and linear acceleration feedback and dynamically neutral stable (limit cycle see [46]) if
The nonlinear frequency content of the statically stable system is given by
and is dynamically stable if
and dynamically unstable if
This extended analysis is equivalent to the concepts of “flight stability” [31, 58] presented in Sect. 4.2, where systems are analyzed and designed based on static stability and dynamic stability. In summary, a system is dynamically stable only if it is also statically stable; a system can be statically stable and dynamically unstable.
4 Lyapunov Analysis
In this section, Lyapunov analysis is utilized to prove the necessary and sufficient conditions for stability of a class of nonlinear systems, Hamiltonian natural systems, based on the concepts of static stability and dynamic stability with respect to the energy storage surface and power flow. In particular, the Hamiltonian surface and its time derivative determine the necessary and sufficient conditions for stability.
-
Necessary:
$$\mathcal{H} (\dot{\mathbf{q}},\mathbf{q}) =\mathcal{T}(\dot{\mathbf{q}},\mathbf{q}) +\mathcal{V}(\mathbf{q})$$ -
Sufficient:
$$\dot{\mathcal{H}}(\dot{\mathbf{q}},\mathbf{q})$$
The proofs will be based on theorems and lemmas from references [6, 14, 52, 59] and [13, 46]. This development is the basis of the HSSPFC control law design and analysis procedure that will be presented and demonstrated in the next section.
Lyapunov analysis is based on the straightforward idea that if the state of a system is near an equilibrium state and the energy of the system is decreasing, then the equilibrium state is stable, possibly asymptotically stable. However, if the energy is increasing near an equilibrium state, then the equilibrium state is unstable. Lyapunov stability, theorems, and functions are an extension and generalization of the energy concept that has been utilized in the previous discussions of stability. The Lyapunov function is the energy storage surface (Hamiltonian) which determines the accessible phase space of the system (constraint surface/manifold). The time derivative of the Lyapunov function is the power flow (work/rate) which determines the path/trajectory of the system as it traverses the Hamiltonian given some initial condition. It will be demonstrated that the design of the Lyapunov function, Hamiltonian Surface Shaping, is equivalent to static stability, which is a necessary condition for stability. The design of the time derivative of the Lyapunov function, Power Flow Control, is equivalent to dynamic stability, which is a sufficient condition for stability for Hamiltonian natural systems.
The proof begins with the stability analysis of conservative dynamical systems which is basically a restatement of the analysis done for static stability within the context of Lyapunov analysis. The system is a conservative natural Hamiltonian system with no externally applied nonconservative forces or moments. The Lagrange–Dirichlet Theorem (a state where the potential energy is an isolated minimum is a stable equilibrium state [52]), which is equivalent to the Lyapunov Stability Theorem [51] (if in some neighborhood Ω of the origin, there exists a Lyapunov function V(x) (where V(0)=0, V(x)>0 ∀x≠0, and \(\dot{V}(x) \leq0\)), then the origin is stable [52]) can be applied at this point to the energy storage surface Lyapunov candidate function, which is a constant, or
The equations of motion are given in first-order canonical form as
and second-order form as
A conservative dynamical system is (statically) stable if the Lyapunov function (Hamiltonian and potential functions) is positive definite about the equilibrium state and its time derivative is zero,
and
and
The Converse of the Lagrange–Dirichlet Theorem, Lyapunov’s Theorem (at an isolated maximum of the potential energy, the equilibrium state is unstable [52]), can be applied, and the system is (statically) unstable if
where the Lyapunov candidate function can be chosen as the Lagrangian
The Instability Theorem of Chetayev [14] or Extended Lyapunov’s Theorem (if at an equilibrium state the potential energy is not a minimum, then the equilibrium state is unstable [52]) can be applied, and the system is unstable (statically neutrally stable) if
Notice that a conservative dynamical system transitions from stable to unstable as the potential energy function is deformed from a positive definite function to a zero function to a negative definite function. The onset of instability occurs at the point where the potential energy function loses its positive definite convexity. This is presented in Fig. 4.16. Also, a conservative dynamical system is precluded from being asymptotically stable since no external damping forces can exist.
Now, it is necessary to determine what effect power flow, damping and generation, have on conservative dynamical systems. The system is modeled as a natural Hamiltonian system with externally applied nonconservative forces and/or moments. The equations of motion are given in first-order canonical form as
and in second-order form as
Chetayev [14] and Meirovitch [6] have investigated this situation in detail for complete damping and pervasive damping. Complete damping occurs when
is a negative definite function of the generalized velocities \(\dot{q}_{j}\). Pervasive damping occurs when \(\dot{\mathcal{H}}\) is a negative semi-definite function of \(\dot{q}_{j}\) and the set of points where \(\dot{\mathcal{H}}=0\) contain no nontrivial positive half-trajectory of the system.
Chetayev [14] and Meirovitch [6] prove: Dissipative forces do not disturb stability of the equilibrium state of a conservative dynamical system in a meaningful way (static stability is a necessary condition for stability).
-
1.
If the equilibrium state is stable with potential forces, it becomes asymptotically stable with the addition of dissipative forces with complete damping.
-
2.
An equilibrium state which is unstable with potential forces cannot be stabilized by dissipative forces.
Two general theorems follow from these investigations:
Theorem 4.1
[6]
If for the system of equations (4.67) and (4.68), the Hamiltonian is positive definite and if the system possesses pervasive damping (complete damping as a subset), then the equilibrium state is asymptotically stable.
Theorem 4.2
[6]
If for the system of equations (4.67) and (4.68), the Hamiltonian can assume negative values in the neighborhood of the origin and if the system possesses pervasive damping (complete damping as a subset), then the equilibrium state is unstable.
Actually, if the potential energy function is \(\mathcal{V} (\mathbf{q}) =0\) ∀q, then the system is unstable with complete damping since the system has no preferred orientation, a rigid body mode and singular stiffness matrix. The way to solve this problem is with a two-step control law design process that first utilizes proportional and/or acceleration feedback to shape the Hamiltonian surface to ensure static stability. The second step ensures dynamic stability via damping and/or generation feedback. This two-step process, HSSPFC, will be discussed in more detail in the next section.
Returning to the discussion of the stability of the system of equations (4.67) and (4.68), the system path/trajectory traverses a positive definite energy storage surface (statically stable) defined by the Hamiltonian as a result of the power flow relative to an initial condition. The time derivative of the energy storage surface defines the power flow into, dissipated within, and stored in the system. This determines whether the system is rising to a higher energy state (away from its equilibrium state), dropping to a lower energy state (returning to its equilibrium state), or staying on a closed cyclic path (limit cycle) constrained to the energy storage surface. Average power flow calculations are used because one cannot guarantee that the opposing power flows will cancel or be dominant generators or dissipators point-for-point in time [46]. In fact, limit cycles balance over the cycle as described by [6]: A state of stationary motion is achieved in which the system gains energy during part of the cycle and dissipates energy during the remaining part, so that at the end of each cycle the net energy exchange is zero. If the system were linear, then the power flows would cancel or be dominant generators or dissipators point-for-point in time [46]. Consequently, feedback linearization has some attractive features when designing feedback controllers.
The Lyapunov analysis of the system of equations (4.67) and (4.68) provides the sufficient conditions for (dynamic) stability. The system is asymptotically (dynamically) stable if [13]
The system is (dynamically) unstable if [13]
The limit cycle defines the stability boundary between asymptotically (dynamically) stable and (dynamically) unstable [46] and occurs when
Notice that the (static) stability boundary for a conservative dynamical system is a rigid body mode where the potential energy function loses its positive definite convexity, while the (dynamic) stability boundary for the system of equations (4.67) and (4.68) is a limit cycle or a second-order center for a linear system. The (static) stability of a conservative dynamical system is a necessary condition for the (dynamic) stability of the system of equations (4.67) and (4.68), which provides the sufficient condition for stability. The necessary and sufficient conditions for stability of a class of nonlinear systems, Hamiltonian natural systems are the basis for a two-step control law design and analysis process called Hamiltonian Surface Shaping and Power Flow Control (HSSPFC).
5 Energy Storage Surface and Power Flow: HSSPFC
The goal of this section is to demonstrate how to design and analyze nonlinear controllers for a class of nonlinear systems, Hamiltonian natural systems from mechanics and adiabatic irreversible work processes from thermodynamics, with a two-step process called Hamiltonian Surface Shaping and Power Flow Control (HSSPFC).
The first example presents the Hamiltonian Surface Shaping of a conservative linear and cubic nonlinear spring system. The bifurcated potential energy surface is given as
for k,k NL>0, and the proportional controller is
that leads to the Hamiltonian
which is statically stable if K P −k>0. The equations of motion become
and the Hamiltonian surface is presented in Figs. 4.17 and 4.18, respectively.
To determine the effect that the proportional controller gain K P has on the system, Hamiltonian phase plane plots are generated. By investigating a system with negative stiffness and by adding enough K P to result in an overall positive net stiffness, the shape of the Hamiltonian surface changes from a saddle-point surface (see Fig. 4.17) to a positive bowl surface (see Fig. 4.18). A two-dimensional cross-section of the Hamiltonian versus the position presents the characteristics of the overall storage or potential functions. The operating point at \((\mathcal{H},\dot{x},x) = (0,0,0)\) changes from being unstable to stable for small values of |x|>0, when enough additional K P is added, a net positive stiffness for the system results.
In example two, Hamiltonian Surface Shaping of a conservative nonlinear inverted pendulum with an externally applied conservative torque is investigated. The potential energy for the nonlinear pendulum hanging straight down is given as
for m,g,l>0 that leads to the Hamiltonian
which leads to the equation of motion
The nonlinear inverted pendulum model can be obtained by a transformation of coordinates as θ=π+β, which leads to
and the Hamiltonian as
and the equation of motion becomes
This is a repulsive potential that is statically unstable. A nonlinear proportional feedback controller of the form
that derives from a potential of the form
then the Hamiltonian becomes
which results in the following equation of motion:
The system can be made statically stable if
The second step in the HSSPFC process is Power Flow Control which shapes the path/trajectory across the energy storage surface. This is accomplished by designing the balance of power flowing into versus the power being dissipated within the system as a function of the power being stored in the system and the initial condition. Power Flow Control is implemented with the derivative feedback (dissipator) and/or integral feedback (generator) portions of the control law. Power flow in the context of mechanics is referred to as exergy flow in the context of thermodynamics and is partitioned into three types: power flowing into (generator), dissipated within (dissipator), and stored in the system (storage). The balance of these power flows determines the path/trajectory across the energy storage surface as a function of the initial condition and the dynamic stability of the system. The time derivative of the Hamiltonian is partitioned into generators, \(\dot{W}\), dissipators, \(T_{0} \dot{S}_{i}\), and storage terms, \(\dot{\mathcal{H}}\), in order to design the power flow balance defined by
Derivative feedback is a dissipator that creates irreversible entropy,
Integral feedback is a generator that flows power into the system,
A few examples are presented to demonstrate Power Flow Control.
Example one is the power flow control of the linearized mass, spring, damper system in Fig. 4.4 with numerical simulation results presented in Figs. 4.5–4.14 where the effects of integral feedback are demonstrated in detail. This system is statically stable while being dynamically stable, neutral stable, and unstable depending upon the feedback gains.
The second example is a Duffing oscillator equation with Coulomb friction given by
and controlled with a PID controller
The Hamiltonian is
with time derivative
The nonlinear limit cycle occurs when
Numerical simulations were performed to demonstrate where the nonlinear stability boundary lies for the Duffing oscillator/Coulomb friction dynamic model subject to PID control. Twelve separate cases (Cases 1–12) were conducted with the numerical values listed in Table 4.2. Cases 1–4 and gain-scheduling of this example are discussed in Sect. 3.6.2 of Chap. 3.
It is instructive to present what happens when the statically stable Hamiltonian surface about the equilibrium point (0,0) is bifurcated into two equilibrium points near the origin, which makes the system statically unstable at the origin. This situation happens when the linear stiffness is negative, K P +k<0, and is presented in Cases 5–12 in Table 4.2. For Cases 5–10, the system is under regulator control, at \((x,\dot{x})=(0,0)\) with positive proportional feedback. This causes the point \((x,\dot{x}) = (0,0)\) to be an unstable static node, and for the following scenarios with K P +k<0, the system is forced away from the regulator point.
In Case 5, the system builds up enough energy to transition out of the right well but overshoots into the left well, where the process starts again (see Figs. 4.19, 4.20, 4.21, and 4.22). Eventually, the system achieves a balanced equilibrium between both wells, a nonlinear limit cycle. A similar response results in Case 6 (see Figs. 4.23, 4.24, 4.25, and 4.26). Note that for a reduction in K I , the responses are slower, in comparison to the previous Case 5.
In Case 7 (see Figs. 4.27 and 4.28), the K I again is reduced resulting in the appearance that the system decays down to a point in the right well. Also note that in Fig. 4.29 the dissipative term is greater than the generator term with corresponding decaying responses in Fig. 4.30 for both the position and velocity, respectively. Case 7 is building up slower than previous cases, since K I is reduced. However, in Cases 7 and 8, given enough simulation time (t f =100 sec), the generator term eventually builds up enough energy to move out of the right well, but again overshoots (see Figs. 4.31 and 4.32) the (0,0) regulator point and spirals down into the left well. The exergy and exergy-rate responses are given in Fig. 4.33 along with the corresponding state responses in Fig. 4.34.
For Case 9, K I is increased such that the system traverses around both left and right wells and approaches another higher energy level, nonlinear limit cycle (see Figs. 4.35 and 4.36 along with Figs. 4.37 and 4.38). In this case the generator term maintains a balance with the opposing damping terms subject to the nonlinear spring effect.
Case 10 demonstrates a reduction in K I and also starts in the left well, builds up enough energy to move over to the right well, and eventually comes back to another stable energy state, nonlinear limit cycle (see Figs. 4.39, 4.40, 4.41, and 4.42).
For the last two cases (Cases 11 and 12), the PID regulator is converted to a tracking controller for which
where for this discussion, x r is a reference step input, and the reference velocity is \(\dot{x}_{r} = 0\). The corresponding Hamiltonian becomes
In Case 11, the system is given a unity reference step input for which after moving from (0,0) the system spirals into and converges into the right well at (1,0) (see Figs. 4.43, 4.44, 4.45, and 4.46).
This HSSPFC analysis and design has provided insight for the investigation of forced nonlinear systems. Both stability and performance can be further characterized and synthesized with a better understanding of limit cycles and their relationship and role played with respect to the nonlinear system.
In Case 12, the system also starts at (0,0) and is commanded to step into the left well at (−1,0) for which the results also spiral and converge to the appropriate final condition (see Figs. 4.47, 4.48, 4.49, and 4.50). In addition, for comparative purposes without simulation plots, the reference operating point was set to \((x_{r}, \dot{x}_{r}) = (0,0)\) (the unstable node), and for nonzero initial conditions, the tracking controller still eventually overshoots the set point with very similar results to the previous cases.
6 Chapter Summary
Chapter 4 presented the concepts of static stability, dynamic stability, eigenanalysis, and Lyapunov analysis that were used to develop the necessary and sufficient conditions for stability of a class of nonlinear systems, Hamiltonian natural systems. The combination of static stability of conservative systems and dynamic stability of adiabatic irreversible work processes in the form of exergy rate equations were used to develop Hamiltonian Surface Shaping and Power Flow Control (HSSPFC). Several examples demonstrated the HSSPFC controller design process that will be applied to a series of case studies in Part II.
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Robinett, R.D., Wilson, D.G. (2011). Stability and Control. In: Nonlinear Power Flow Control Design. Understanding Complex Systems. Springer, London. https://doi.org/10.1007/978-0-85729-823-2_4
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