Unwind di Power Electronics - 4 di 5

Lesson #20 lesson #21 compensator design with the bilinear transform method

Once the uncompensated loop gain T (z) is determined using either the discretization-based approach or the discrete-time approach, one proceeds to designing the discrete-time compensator transfer function D (z).c

• An effective design method which makes use of the bilinear transform will be discussed in the following.
• The end result of the compensator design step is the determination of the numerical values of the proportional, integral and derivative gains of the G (z):c K 1iG ( z ) K K (1 z )−= + + ⋅ −c p d11 z −−

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The Transfer Function (TF) Tu(z) represents small signal model of the digitally controlled converter provided by the discretisation base model approach. This TF describes (in a small signal way) the relation between the modulating signal u and the sensed signal xs (this last signal can be either a voltage or a current signal).

The goal is to describe Gc(z) using the bilinear transformation. The general form of the discrete-time PID is: G E1KdZ kptki.it1 z

The bilinear transform design approach

• Step 1: The system – both G (z) and T (z) – is converted into an “analog” equivalent via c u the map: T1 p s+ 2z ( p ) , p C= ∈T1 p s− 2 which has the same structure as the inverse Tustin transform.
• Step 2: The compensator G' (p)=G (z(p)) is designed in the “continuous-time” c cp-plane via analog control design techniques.
• Step 3: The compensator is brought back to the z-plane by application of the inverse map.
• Remark: Contrary to continuous-time methodologies based on controller discretization, the bilinear transform method does not introduce approximations.

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This approach is divided in 3 steps.

In the first step, the z-domain expressions of the uncompensated loop gain Tu(z) and the transfer function of the compensator Gc(z) into an equivalent analog domain, characterised with the variable p. The equivalent analog TFs are defined in a p-plane. To realise this transformation it is used the bilinear transformation, defined as follow:

The variable p acts like a complex frequency.

Lets see now the z-plane and p-plane:

What happens thanks to this transformation is the following: whatever is described inside the unitary circle in the z-plane is mapped in the left half plane of the p-plane, whatever is described outside the unitary circle in the z-plane is mapped in the right half plane of the p-plane and the unit circle is mapped on the imaginary axis.

Note that the inside portion of the unitary circle is the stable portion of the z-domain and this portion is represented into the left side of the p-domain. The left side of the imaginary axis is usually associated to the stable portion of the plane also in the Laplace domain. Similarly for unstable portions of the z-domain and of the p-domain.

The actuated transformation so maps the z-domain in an equivalent complex plane used to characterize a continuous time (analog) system. The transformation is simply realised substituting the variable z with this expression. The expression is also invertible, so if p is expressed as function of z, the analysis moves from the p-plane to the z-plane.

In the second step it’s designed a compensator that is: Gc’(p)=Gc(z(p)), and it’s designed into the analog p-plane, so it must be used all the rules defined in continuous time control designed. The big advantage of this approach is so that the design happens in an equivalent continuous time environment.

In the last step of the bilinear approach there is the application of the inverse of the bilinear transform that simply transforms Gc’(p) into Gc(z).

This approach doesn’t introduce any approximation or any frequency domain distortion, because any type of distortion introduced in the first step (moving from the z-plane to the p-plane) is cancelled in the last step (moving from the p-plane to the z-plane).

Basic properties of the bilinear transform

• If G (z) is rational in the variable z, then G' (p) is rational in the variable p c c
• The unit disk |z|<1 is mapped into the left half-plane Re[p]<0, while the unstable portion of the z-plane |z|>1 is mapped into the right half-plane Re[p]>0
• The unit circle in the z-plane is mapped into the imaginary axis of the p-plane
• G' (p) can be interpreted and analyzed as the transfer function of a continuous-time system, and G' (j ) as its frequency response. ωc

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Design flow using the bilinear transform

z-domain p-domain “Continuous-Time” pT1Start: Uncompensated loop s+ equivalent T' (p) + specs2 uz ( p) =gain T (z) + specifications pTu 1 s− 2 Conventional analog design End: Discrete-Time PID 12 1 z − “Continuous-Time” PID−p ( z )compensator G (z) = 1c T 1 z − compensator G' (p)+s cK 1iG ( z ) K K (1 z )−= + + ⋅ −c p d11 z −−

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Here is represented an example of flow-chart that can be used to achieve the bilinear transformation. Note that the uncompensated loop gain Tu(z) and the z-domain specifications are both converted in the p-domain in order to achieve the definition of a compensator TF Gc’(p). At least, this p-domain TF is converted in the z-domain using p(z), defined as follow:

Example: Design of a digital PI compensator

• Suppose T' (p) has been derived from the bilinear transform of T (z). u u
• Goal is to synthesize a digital PI compensator from frequency-domain specifications expressed in terms of control bandwidth and phase margin .ω ϕc m K
• In the z-domain the PI compensator has the usual form iG ( z ) K = +c p 11 z −−
• Step 1: In the p–plane the PI is written as: pT1 s+  K ω2  i PIG ' ( p ) K G 1= + = + c p PI∞T p p sK 2ω piG K , ,with ω ω= + = =PI p PI p2 K∞ 2 Tp1 s+ K i

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Suppose T'u(p) has been derived from the bilinear transform of Tu(z) and the desired Gc(z) is:

The corresponding p-domain compensator TF is obtained simply substituting the variable z with it’s definition as function of p:

Where:

Example: Design of a digital PI compensator

• Step 2: Design of G' (p) proceeds by first imposing the phase margin constraint, c which determines the position of the PI zero:  ω G ' ( j ) T ' ( j ) PIarctan T ' ( j )ω ω π ϕ π ω ϕ ϕ ϕ∠ + ∠ = − + − = − − ∠ + = − c c u c m u c m m m , uω c where "uncompensated phase margin"T ' ( j )ϕ π ω= + ∠m , u u c( )
• Solving for : tan ω ω ω ϕ ϕ= −PI PI c m , u m
• The control bandwidth constraint yields the value of the integral gain: 1G ' ( j ) T ' ( j ) 1 Gω ω⋅ = =c c u c PI∞ 2 ω T ' ( j ) 1 PIω ⋅ +  u c  ω c

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At this point it must simply applied the project specifications in order to determine the values of GPIinf and of wPI. Using the constraint on the phase yields the following result:

While the constraint on the control bandwidth yields the following result:

Example: Design of a digital PI compensator

• Step 3: With G and so determined, compensator parameters in the z-plane are ωPI∞ PI found by inverting equations  K ω PIK G 1iG K = −= +  PI p p PI∞ ∞2 ω pω 2p ωω = PIK GPI 2 K =i PIp ∞1 ω+ pK i
• Note: observe that a valid PI solution K exists if and only if 0 < , i.e. >0 < ω ωp,i PI p ω parctanϕ ϕ ϕ− < < m , u m m , u ω c A PI compensation cannot boost the phase margin!

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The expressions of Kp and Ki are:

At this point, the design of Gc(z) is completed because all it’s parameters are obtained.

Note that to obtain a valid PI solution, it must be satisfied the following condition: Kp,i>0, which yields another result that is: 0<wPI<wp. This expression yields another constraint on the phase margin. In fact if wPI must be greater then 0, the following expression:

Yields that:

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The desired phase margin cannot be greater the the uncompensated phase margin.

Note that in all the calculations has been used the cross-over frequency wc that is a z-domain specification that has not been converted in a p-domain specification. This yields a frequency distortion that can be compensated with an operation called prewarping.

Prewarping

• In the previous example, the crossover frequency expresses the desired control ωc bandwidth in the discrete domain. Therefore, frequency axis distortion introduced by the bilinear transform has been neglected.
• To compensate the frequency distortion, one simply has to transform the control bandwidth specification into a corresponding specification ' in the p-plane, an ω ωc c operation usually referred to as prewarping:  ω c' tanω ω=  c p ω p
• Compensator design then proceeds as before, with ' replacing . ω ωc c1 /ω0.95 ωc c,