generalized modal control
Reminder: Before reading this article, readers should have certain modal knowledge, which can be learned by reading Section 1 of the author’s other blog modal feedback control .
1. The idea of mode and mode control
We know that the time domain motion of the system can be expressed by the following equation:
x ( t ) = ∑ i = 1 n C ie λ it = ∑ i = 1 nxi ( t ) x(t) = \sum _{i=1} ^n C_i e^{\lambda_i t} = \sum_{i=1} ^n x_i (t)x(t)=i=1∑nCielit=i=1∑nxi( t ) λ ite^{\lambda_i t}elit is the poleλ i \lambda_iliThe corresponding movement mode.
The modal control is to find a way to add the desired mode to the system, or eliminate the bad mode in the system, so that the system can achieve the desired dynamic characteristics .
2. Mathematical derivation of control method
Suppose the characteristic equation of the original system is
Δ 0 ( s ) = det ( s I − A ) = sn + ∑ i = 1 n − 1 aisi (1) \Delta_0 ({\rm s} ) = \det \left( {\rm s} I - A \right) = {\rm s}^n + \sum_{i=1} ^{n-1} a_i {\rm s}^i \tag{1}D0(s)=the( s I−A)=sn+i=1∑n−1aisi( 1 ) Note: Formula (1) is the, that is,the coefficient of the first term (the term with the highest power) is 1.
Correspondingly, the transfer function of the system is
W 0 ( s ) = CT ( s I − A ) − 1 B = M 0 ( s ) Δ 0 ( s ) W_0 ( {\rm s} ) = C^{\rm T } \left( {\rm s} I - A \right)^{-1} B = \frac{M_0 ( {\rm s} )}{\Delta_0 ({\rm s} )}W0(s)=CT( s I−A)−1B=D0(s)M0(s)The denominator is shown in formula (1), and the numerator is
M 0 ( s ) = ∑ i = 1 mbisi M_0 ( {\rm s} ) = \sum_{i=1} ^m b_i {\rm s}^iM0(s)=i=1∑mbisiGenerally , for practical systems, the order of the numerator will not exceed the order of the denominator, that is,m ≤ nm \leq nm≤n。
In the forward path of the system, add a controller R ( s ) = Q ( s ) / P ( s ) R ( {\rm s} ) = Q ( {\rm s} ) / P ( {\rm s } )R(s)=Q ( s ) / P ( s ) , its order isν \nuν，即deg P ( s ) = ν \deg P ( {\rm s} ) = \nowdegP(s)=v . For practical systems, there is also a numerator order that does not exceed the denominator order, that is,deg Q ( s ) ≤ deg P ( s ) \deg Q ( {\rm s} ) \leq \deg P ( {\rm s} )degQ(s)≤degP(s)。
The controller R ( s ) R ( {\rm s} )R(s)表示为
R ( s ) = Q ( s ) P ( s ) = ∑ i = 0 ν q i s i s ν + ∑ i = 0 ν − 1 p i s i (2) R ( {\rm s} ) = \frac{Q ( {\rm s} )}{P ( {\rm s} )} = \frac{ \sum_{i=0} ^\nu q_i {\rm s}^i }{ {\rm s}^\nu + \sum_{i=0} ^{\nu-1} p_i {\rm s}^i } \tag{2} R(s)=P(s)Q(s)=sn+∑i=0n − 1pisi∑i=0nqisi( 2 ) Note that the denominator PPhereP is also the first formula.
After adding the controller to the forward path, the closed-loop transfer function of the whole new system can be written immediately:
W ( s ) = R ( s ) W 0 ( s ) 1 + R ( s ) W 0 ( s ) = M ( s ) Δ ( s ) = Q ( s ) P ( s ) M 0 ( s ) Δ 0 ( s ) 1 + Q ( s ) P ( s ) M 0 ( s ) Δ 0 ( s ) = Q ( s ) M 0 ( s ) P ( s ) Δ 0 ( s ) + Q ( s ) M 0 ( s ) (3) \begin{aligned} W ( {\rm s} ) &= \frac{ R ( {\rm s } ) W_0 ( {\rm s} ) }{1 + R ( {\rm s} ) W_0 ( {\rm s} ) } = \frac{M ( {\rm s} ) }{ \Delta ( {\ rm s} ) } \\ &= \frac{ \frac{Q ( {\rm s} )}{P ( {\rm s} )} \frac{M_0 ( {\rm s} )}{\Delta_0 ( {\rm s} )} }{1 + \frac{Q ( {\rm s} )}{P ( {\rm s} )} \frac{M_0 ( {\rm s} )}{\Delta_0 ({ \rm s} )} } \\ &= \frac{Q( {\rm s} ) M_0( {\rm s} )}{P( {\rm s} ) \Delta_0( {\rm s} ) + Q( {\rm s} ) M_0( {\rm s} )} \tag{3} \end{aligned}W(s)=1+R(s)W0(s)R(s)W0(s)=D ( s )M(s)=1+P(s)Q(s)D0(s)M0(s)P(s)Q(s)D0(s)M0(s)=P ( s ) D0(s)+Q(s)M0(s)Q(s)M0(s)(3)即
Δ ( s ) = P ( s ) Δ 0 ( s ) + Q ( s ) M 0 ( s ) , (3–1) \Delta ( {\rm s} ) = P( {\rm s} ) \Delta_0( {\rm s} ) +Q( {\rm s} ) M_0( {\rm s} ), \tag{3--1} D ( s )=P ( s ) D0(s)+Q(s)M0(s),(3–1) M ( s ) = Q ( s ) M 0 ( s ) (3–2) M ( {\rm s} ) = Q( {\rm s} ) M_0( {\rm s} ) \tag{3--2} M(s)=Q(s)M0(s)( 3–2 ) Let’s look at the order of formulas (3–1) and (3–2).
For formula (3–1):
deg Δ ( s ) = max { deg P ( s ) + deg Δ 0 ( s ) , deg Q ( s ) + deg M 0 ( s ) } \deg \Delta ( {\rm s} ) = \max \left\{ \deg P( {\rm s} ) + \deg \Delta_0( {\rm s} ), \quad \deg Q( {\rm s} ) + \deg M_0( {\rm s} ) \right\}degD ( s )=max{
degP(s)+degD0(s),degQ(s)+degM0( s ) } sincedeg P ( s ) = ν , deg Δ 0 ( s ) = n \deg P( {\rm s} ) = \nu, \deg \Delta_0( {\rm s} ) = ndegP(s)=n ,degD0(s)=n，而 Q ( s ) Q( {\rm s} ) Q(s)与 M 0 ( s ) M_0( {\rm s} ) M0( s ) are all molecules, and their order does not exceedν \nun与nnn，故
deg Δ ( s ) = n + ν = n ˉ (4) \deg \Delta ( {\rm s} ) = n + \nu = \bar n \tag{4}degD ( s )=n+n=nˉ( 4 )
Letthe characteristic equation of thedesiredΔ ∗ ( s ) \Delta ^* ( {\rm s} )D∗ (s), then only need to satisfy
Δ ∗ ( s ) = Δ ( s ) = P ( s ) Δ 0 ( s ) + Q ( s ) M 0 ( s ) (5) \Delta ^* ( {\rm s} ) = \Delta ( {\rm s} ) = P( {\rm s} ) \Delta_0( {\rm s} ) +Q( {\rm s} ) M_0( {\rm s} ) \tag {5}D∗(s)=D ( s )=P ( s ) D0(s)+Q(s)M0(s)( 5 ) can be. Solving formula (5), you can get the numerator and denominator of the controllerP ( s ) , Q ( s ) P( {\rm s} ), Q( {\rm s} )P(s),Q(s)。
3. Several conditions for the solution
- The necessary and sufficient condition for the existence of the solution of formula (5) is: Δ 0 ( s ) \Delta_0( {\rm s} )D0( s ) givenM 0 ( s ) M_0( {\rm s} )M0( s ) coprime, andn ˉ ≥ 2 n − 1 \bar n \geq 2n - 1nˉ≥2 n−1。
- 当2 ν + 1 ≥ n ˉ 2\nu + 1 \geq \bar n2 n+1≥nˉ , formula (5) has infinite solutions; only when2 ν + 1 = n ˉ 2\nu + 1 = \bar n2 n+1=nˉ , formula (5) has a unique solution.
- It can be known from 1 that if and only if ν ≥ n − 1 \nu \geq n-1n≥n−1 , only by setting the controllerR ( s ) R ( {\rm s} )R ( s ) makes the system obtain the desired pole distribution. Therefore, the controllerR ( s ) R ( {\rm s} )The minimum order of R ( s ) ν 0 = n − 1 \nu_0 = n-1n0=n−1。
4. Example calculation
Suppose the transfer function of the original system is
W 0 ( s ) = k 0 s ( T s + 1 ) W_0 ({\rm s}) = \frac{k_0}{ {
\rm s} \left( T{\rm s } + 1 \right) }W0(s)=s(Ts+1)k0Design a modal controller for it.
Write down its numerator and denominator (note that it is the first formula ):
Δ 0 ( s ) = s ( s + 1 T ) , M 0 ( s ) = k 0 T \Delta_0 ( {\rm s} ) = {\rm s} \left( {\rm s} + \frac{1}{T} \right), \qquad M_0 ( {\rm s} ) = \frac{k_0}{T}D0(s)=s(s+T1),M0(s)=Tk0The original system order n = 2 n=2n=2 , then the minimum order of the controller isν 0 = n − 1 = 1 \nu_0 = n-1 = 1n0=n−1=1 . So the total order isn ˉ = n + ν = 3 \bar n = n + \nu = 3nˉ=n+n=3 , that is, there are 3 expected poles in the new system.
Suppose the 3 expected poles of the new system are λ 1 ∗ = − δ , λ 2 ∗ = λ 3 ∗ = − 2 δ \lambda_1^* = -\delta, \lambda_2^* = \lambda_3^* = -2 \deltal1∗=− d ,l2∗=l3∗=− 2 δ (the expected pole is artificially designed and carried out according to the specific requirements of the system, rather than calculated according to the performance of the system). Then the characteristic equation of the expected system is
Δ ∗ ( s ) = ( s + δ ) ( s + 2 δ ) 2 \Delta^* ( {\rm s} ) = \left( {\rm s} + \delta \right ) \left( {\rm s} + 2\delta \right)^2D∗(s)=(s+d )(s+2 d )2 On the other hand, since the controller of the system is first-order (ν 0 = 1 \nu_0 = 1n0=1 ), so we can set
P ( s ) = s + p 0 , Q ( s ) = q 1 s + q 0 P ( {\rm s} ) = {\rm s} + p_0, \qquad Q ( {\ rm s} ) = q_1 {\rm s} + q_0P(s)=s+p0,Q(s)=q1s+q0Note that P ( s ) P ( {\rm s} )P ( s ) is the first formula.
Definition(5)
Δ ∗ ( s ) = Δ ( s ) = P ( s ) Δ 0 ( s ) + Q ( s ) M 0 ( s ) ⟹ \Delta ^* ( {\rm s} ) = \ Delta ( {\rms} ) = P( {\rms}) \Delta_0({\rms}) +Q({\rms}) M_0({\rms})\LongrightarrowD∗(s)=D ( s )=P ( s ) D0(s)+Q(s)M0(s)⟹ s ( s + 1 T ) ( s + p 0 ) + k 0 T ( q 1 s + q 0 ) = ( s + δ ) ( s + 2 δ ) 2 {\rm s} \left( {\rm s} + \frac{1}{T} \right) \left( {\rm s} + p_0 \right) + \frac{k_0}{T} \left( q_1 {\rm s} + q_0 \right) = \left( {\rm s} + \delta \right) \left( {\rm s} + 2\delta \right)^2 s(s+T1)(s+p0)+Tk0(q1s+q0)=(s+d )(s+2 d )2 Expand the brackets, corresponding tos {\rm s}The coefficients of different orders of s
can be obtained { p 0 + 1 T = 5 δ p 0 T + k 0 T q 1 = 8 δ 2 k 0 q 0 T = 4 δ 3 \begin{cases} p_0 + \frac{ 1}{T} = 5 \delta \\ \frac{p_0}{T} + \frac{k_0}{T} q_1 = 8 \delta^2 \\ \frac{k_0 q_0}{T} = 4 \delta ^3 \end{cases}⎩
⎨
⎧p0+T1=5 dTp0+Tk0q1=8 d2Tk0q0=4 d3Definition
{ p 0 = 5 δ − 1 T q 1 = 8 δ 2 T k 0 − 5 δ k 0 + 1 k 0 T q 0 = 4 δ 3 T k 0 \begin{cases} p_0 = 5\delta - \frac{1}{T}\\q_1 = \frac{8\delta^2 T}{k_0} - \frac{5\delta}{k_0} + \frac{1}{k_0T}\\q_0 = \ frac{4\delta^3 T}{k_0}\end{cases}⎩
⎨
⎧p0=5 d−T1q1=k08 d2 T−k05 d+k0T1q0=k04 d3 TThen the transfer function of the controller of the system is
R ( s ) = Q ( s ) P ( s ) = ( 8 δ 2 T k 0 − 5 δ k 0 + 1 k 0 T ) s + 4 δ 3 T k 0 s + 5 δ − 1 T = ( 8 δ 2 T 2 − 5 δ T + 1 ) s + 4 δ 3 T 2 k 0 T s + 5 δ k 0 T − k 0 R ( {\rm s} ) = \ frac{Q ( {\rm s} )}{P ( {\rm s} )} = \frac{ \left( \frac{8 \delta^2 T}{k_0} - \frac{5\delta}{ k_0} + \frac{1}{k_0T} \right) {\rm s} + \frac{4 \delta^3 T}{k_0} }{ {\rm s} + 5\delta - \frac
{1} {T} } = \frac{ \left( 8 \delta^2 T^2 - 5\delta T + 1 \right) {\rm s} + 4 \delta^3 T^2 }{ k_0 T {\rm s} + 5\delta k_0 T - k_0 }R(s)=P(s)Q(s)=s+5 d−T1(k08 d2 T−k05 d+k0T1)s+k04 d3 T=k0Ts+5δk0T−k0( 8 d2T _2−5δT+1)s+4 d3T _2