Lemma 8.1: Consider two signals e and φ related by the following dynamic equation
8.4 Adaptive Control of Linear Systems with Output Feedback
Consider Lyapunov function candidate
a
where both Γ and P are symmetric positive constant matrix, and P satisfies
Q P A
PA+ T =− Q=QT>0
for a chosen Q . The derivative V& can be computed easily as
a
Therefore, the adaptation law
x
8.4 Adaptive Control of Linear Systems with Output Feedback
Consider the linear time-invariant system presented bu the transfer function
n
where k is called the high-frequency gain. The reason for p this term is that the plant frequency response at high frequency verifies knpm
j
W = −
ω) ω
( , i.e., the high frequency response is essentially determined byk . The relative degree r of this p system is r=n−m. In our adaptive control problem, the coefficients a ,ibj (i=0,1,K,n−1;j=0,1,K,m−1) and the high frequency gain k are all assumed to be unknown. p The desired performance is assumed to be described by a reference model with transfer function
m
theory that the relative degree of the reference model has to be larger or equal to that the plant in order to allow the possibility of perfect tracking. Therefore, in our treatment, we will assume that nm−mm≥n−m.
The objective of the design is to determine a control law, and an associated adaptation law, so that the plant output y asymptotically approachesy . We assume as follows m
- the plant order n is known - the relative degree n− is known m - the sign of k is known p
- the plant is minimum phase
8.4.1 Linear systems with relative degree one Choice of the control law
To determine the appropriate control law for the adaptive controller, we must first know what control law can achieve perfect tracking when the plant parameters are perfect known.
Many controller structures can be used for this purpose. The following one is particularly convenient for later adaptation design.
Example 8.5 A controller for perfect tracking_____________
Consider the plant described by
a u
and the reference model
a r
Fig. 8.13 Model-reference control system for relative degree 1 Let the controller be chosen as shown in Fig. 8.13, with the control law being
r
function from the reference input r to the plant output is
)
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Chapter 8 Adaptive Control 42
Therefore, perfect tracking is achieved with this control law, i.e., 0y(t)=ym(t),∀t≥ .
Why the closed-loop transfer function can become exactly the same as that of the reference model ? To know this, note that the control input in (8.47) is composed of three parts:
- The first part in effect replaces the plant zero by the reference model zero, since the transfer function from u to 1
y is
- The second part places the closed-loop poles at locations of those of reference model. This is seen by noting that the transfer function from u to y is 0 a result of the above three parts, the closed-loop system has the desired transfer function.
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The above controller in Fig. 8.13 can be extended to any plant with relative degree one. The resulting structure of the control system is shown in the Fig. 8.14, where k*,θ1*,θ*2 andθ *0 represents controller parameters which lead to perfect tracking when the plant parameters are known.
e
Fig. 8.14 A control system with perfect tracking The structure of this control system can be described as follows:
- The block for generating the filter signalω represent an 1 n 1)th
( − order dynamics, which can be described by hu intended to modulate the high frequency gain of the control system.
- The vector θ1*contains ( −n 1)parameters which intend to cancel the zeros of plant.
- The vector θ*2contains )(n−1 parameters which, together with the scalar gain θ0* can move the poles of the closed-loop control system to the locations of the reference model poles.
As before, the control input in this system is a linear combination of:
- the reference signalr(t)
- the vector signalω obtained by filtering the control input u 1 - the vector signalω obtained by filtering the plant output y 2 and the output itself.
The control input can be rewritten in terms of the adjustable parameters and the various signals, as
y
Corresponding to this control law and any reference inputr(t), the output of the plant is
)
since these parameters result in perfect tracking. At this point, we can see the reason for assuming the plant to be minimum-phase: this allows the plant zeros to be cancelled by the controller poles.
In adaptive control problem, the plant parameters are unknown, and the ideal control parameters described above are also unknown. Instead (8.49), the control law is chosen to be
y provided by the adaptation law.
Choice of adaptation law
For the sake of simplicity, define as follows
[
kt t t t]
TThen the control law (8.51) becomes ) (8.52), the control system with variable gains can be equivalently represented as shown in Fig. 8.15, withφT(t)ω/k* regarded as an external signal. The output
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Chapter 8 Adaptive Control 43
)
Fig. 8.15 An equivalent control system for time-varying gains Sinceym(t)=Wm(p)r, the tracking error is seen to be related to the parameter error by the simple equation
]
Since this is the familiar equation seen in Lemma 8.1, the following adaptation law is chosen
)
whereγ is positive number representing the adaptation gain and we have used the fact that the sign ofk is the same as that * ofk , due to the assumed positiveness ofp k . m
Based on Lemma 8.1 and through a straightforward procedure for establishing signal boundedness, we can show that the tracking error in the above adaptive control system converges to zero asymptotically.
8.4.2. Linear system with higher relative degree
The design of adaptive controller for plants with relative degree larger than 1 is both similar to, and different from, that for plants with relative degree 1. Specifically, the choice of control law is quite similar but the choice of adaptation law is very different.
Choice of control law
Let us start from a simple example.
Example 8.6 _______________________________________
Consider the second-order plant described by the transfer function
and the reference model
a r
Fig. 8.16 Model-reference control system for relative degree 2
Let the controller be chosen as shown in Fig. 8.16. Noting that b in the filter in Fig. 8.13 has been replaced by a positive m
number λ. The closed-loop transfer function from the reference signal r to the plant output y is
) chosen such that
) W becomes identically the same as that of the reference ry
model. Clearly, such choice of parameters exists and is unique.
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For a general plants of relative degree larger than 1, the same control structure as given in Fig. 8.14 is chosen. Note that the order of the filters in the control law is still(n−1). However, since the model numerator polynomialZm( p)is of degree smaller than(n−1), it is no longer possible to choose the poles of the filters in the controller so thatdet[pI− Λ]=Zm(p)as in (8.48). Instead, we now choose the reference model can be imposed.
Let us define the transfer function of the feed-forward partu/ u1of the controller byλ(p)/(λ(p)+C(p)), and that of the feedback part byD(p)/λ(p), where the polynomialC( p) contains the parameter in the vector θ , and the 1 polynomialD( p) contains the parameter in the vectorθ . 2 Then the closed-loop transfer function is easily found to be
)
The question now is whether in this general case, there exists choice of values for k,θ1,θ2 andθ such that the above 0
The answer to this question can be obtained from the following lemma
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Chapter 8 Adaptive Control 44
Lemma 8.2: Let A( p) and B( p) be polynomials of
This lemma can be used straight forward to answer our question regarding to (8.59).
Choice of adaptation law
When the plant parameters are unknown, the controller (8.52) is used again
and the tracking error from (8.54) ]
However, the adaptation law (8.55) cannot be used. A famous technique called error augmentation can be used to avoid the difficulty in finding an adaptation law for (8.62). The basic idea of the technique is to consider a so-called augmented errorε(t)which correlates to the parameter error φ in a more desirable way than the tracking errore(t).
)
Fig. 8.17 The augmented error Let define an auxiliary errorη(t)by
as shown in Fig. 8.17. It is useful to note two features about this error
- Firstly, )η(t can be computed on-line, since the estimated parameter vectorθ(t) and the signal vectorω(t) are both available.
- Secondly, η(t)is caused by time-varying nature of the estimated parametersθ(t), in the sense that whenθ(t)is replaced by the true (constant) parameter vectorθ , we * have θ*Wm(p)[ω(t)]−Wm(p)[θ*(t)ω(t)]=0. This also implies thatηcan be written:η(t)=φTWm(ω)−Wm(φTω) Define an augmented errorε(t)
)
whereα(t)is a time-varying parameter to be determined by adaptation. For convenience, let us writeα(t)in the form
)
This implies that the augmented error can be linearly parameterized by the parameter errorφ(t)andφα. Using the gradient method with normilazation, the controller parameters
) (t
θ and the parameterα(t)for forming the augmented error are updated by
ω
With the control law (8.61) and adaptation law (8.67), global convergence of the tracking error can be shown.