Internal stability

Stability is one of the most important objectives of (linear and nonlinear) control systems design. There exists two approaches in the analysis of a dynamic system stability, known as Lyapunov stability (including asymptotic stability and exponential stability), and input-output stability (Antsaklis & Michel, 1997). In Lyapunov stability, the deviation of the system states from their desired operating points (equilibrium points), in case of applying an external disturbance, is analysed. Input-output stability is another approach to stability investigation (usually for linear systems), which takes system inputs and outputs into consideration. In an input-output stable system, it is expected that every bounded system input will produce a bounded system output. A signal ݑ(ݐ) is defined to be bounded if there exist a constant ܿ such that |ݑ(ݐ)| < ܿ for all ݐ. System properties of this type are referred to as BIBO stability. BIBO stability is important for control systems such as tracking control, where the output of the system is expected to follow a desired input (Antsaklis & Michel, 1997). In this report, we only discuss the criterion and characteristics of BIBO stability condition, which is referred herein after as stability.

The Nyquist’s stability criterion is one of the most common tests to measure the closeness of a linear system to stability:

 According to Nyquist’s stability criterion, the closed loop system is stable if and only if the net number of clockwise encirclements of the point −1 + ݆0 by the Nyquist diagram ܮ(݆߱) = ܩ(݆߱)ܭ(݆߱) plus the number of poles of ܮ(݆߱) in the RHP is zero. For open-loop stable systems ∠ܮ(݆߱) falls with frequency such that∠ܮ(݆߱) crosses −180° only once, as shown

in Figure 4-4, a. The Nyquist stability criterion can also be expressed by logarithmic plot (Bode plot) as follows: the closed-loop system is stable if and only if the loop gain ȁܮȁ is less than 1 at frequency −180° (see Figure 4-4, b) (Ogata, 2010).

Figure 4-4: Typical plot for stable plant; a) Nyquist plot, b) Bode plot.

To test for stability of a feedback system, it is usual to employ stability criteria only for the system input-output transfer function (i.e. from ݎ to ݕ as shown in Figure 4-3), so called “external stability”. However, this assumes that there was no internal RHP pole-zero cancellation between the controller and the plant.

Definition 1 (Doyle, Francis, & Tannenbaum, 1992). A closed-loop system is “internally stable” if none of its components contain hidden unstable modes and the injection of bounded external signals at any place in the closed-loop system results in bounded output signals measured anywhere else in the closed-loop system.

To investigate the internal stability of a closed-loop system, consider a negative feedback control loop as shown in Figure 4-3. By supposing that the output of the plant, the controller and the sensor are linear functions of the sums (or difference) of their inputs:

ݕ = ܩ(݀ + ݑ) ݒ = ܨ(݊ + ݕ)

ݑ = ܭ(ݎ− ݒ) (4-9)

The closed loop control system is called well-posed if all the nine transfer functions from the three exogenous inputs to all internal signals, namelyݑ, ݕ, ݒ and the outputs of the summing junctions, namely ݁, ݔ_ଶ,ݔ_ଷ are exist (Doyle, Francis, & Tannenbaum, 1992). Write the equations at the summing junctions as labelled in Figure 4-3:

݁ = ݎ− ܨݔଷ ݔଶ= ݀ + ܭݔଵ ݔଷ = ݊ + ܩݔଶ

(4-10)

In matrix form these are:

൥−ܭ1 01 ܨ0 0 −ܩ 1൩൥ ݁ ݔଶ ݔଷ ൩= ቈ݀ݎ ݊቉ (4-11)

Thus, the system is well-posed if the above 3x3 matrix is non-singular, that is the determinant 1 + ܩܭܨ is not identically equal to zero. Then the nine transfer functions are obtained from the equation

൥ݔ݁ଶ ݔଷ ൩= ൥−ܭ1 01 ܨ0 0 −ܩ 1൩ ିଵ ቈ݀ݎ ݊቉ = _{1 + ܩܭܨ ൥}1 ܭ1 −ܩܨ −ܨ1 −ܭܨ ܩܭ ܩ 1 ൩ቈ ݎ ݀ ݊቉ (4-12)

If the nine transfer functions in Eq. (4-12) are stable, then it guarantees bounded internal signals for all bounded exogenous signals (BIBO stable) and from definition 1, it is concluded that the feedback system is internally stable. Therefore, for a closed loop control system to be internally stable, not only the system input-output transfer function, i.e. from ݎ to ݕ, should be stable, but also all the internal signals should be bounded for all bounded exogenous signals. In other words, an internally stable system is always externally stable, but not conversely.

To test the internal stability in a simpler way, one can writeܩ, ܭ and ܨ as ratios of coprime factorisations (i.e. polynomials with no common factors):

ܩ =_ܯܰீ ீ , ܭ = ܰ௄ ܯ௄ , ܨ = ܰி ܯி . (4-13)

The characteristic polynomial of the feedback system (i.e. 1 + ܩܭܨ) is the one found by taking the product of the three numerators plus the product of the three denominators:

ܰீܰ௄ܰி+ܯீܯ௄ܯி (4-14)

The zeros of the characteristic polynomials are the closed-loop poles, as seen from(4-12).

Theorem 1 the feedback system is internally stable if there are no closed-loop poles in RHP.

Proof: see (Doyle, Francis, & Tannenbaum, 1992).

□

Therefore, by Theorem 1, internal stability can be determined by checking the zeros of polynomial(4-14).

The requirement of internal stability in a feedback system leads the following statements (Youla, et al., 1974):

1- If ܩ(ݏ) has a RHP-zero at ݖ, then ܮ = ܩܭ, ܶ = ܩܭ/(1 + ܩܭ) and ܩܵ =

ܩ/(1 + ܩܭ), will each have a RHP-zero at ݖ.

2- If ܩ(ݏ) has a RHP-pole at ݌, then ܮ = ܩܭ also have a RHP-pole at ݌,while

ܵ = 1/(1 + ܩܭ) and ܻ = ܭܵ = ܭ/(1 + ܩܭ), will have a RHP-zero at ݌.

ܮ, ܶ, ܵ and ܻ are so called the open loop, the closed loop, the sensitivity and the Youla (parameter) transfer functions, respectively.

Finally, from the above statements, the so called ‘interpolation condition’ could be derived:

If the plantܩ(ݏ) has a RHP-zero ݖor a RHP-pole ݌:

1/ܩ(݌) = 0 ⇒ ܮ(݌) = ∞ ⟺ ܶ(݌) = 1, ܵ(݌) = 0 (4-16)

In general, if the plant has ߲݌ numbers of repeated poles ݌ ,then the interpolation conditions are:

ܶ(݌) = 1, ܵ(݌) = 0 ܽ݊݀ ݀௞ܶ ݀ݏ௞(݌) =

݀௞_ܵ

݀ݏ௞(݌) = 0 1 ≤ ݇ ≤ ߲݌− 1 (4-17) Similarly, if the plant has ߲ݖ numbers of repeated zeros, then the interpolation conditions are (Assadian F., 2011):

ܶ(ݖ) = 0, ܵ(ݖ) = 1 ܽ݊݀ ݀_݀ݏ௞ܶ_௞(ݖ) =݀_݀ݏ௞ܵ_௞(ݖ) = 0 1 ≤ ݇ ≤ ߲ݖ− 1 (4-18)

The conditions clearly restrict the allowable ܵ and ܶ to achieve internal stability and also could be used as a measure for verifying internal stability of the system.