[<< wikibooks] LMIs in Control/pages/LMI for L2-Optimal State-Feedback Control under Time-Varying Input Delay
LMIs in Control/pages/LMI for L2-Optimal State-Feedback Control under Time-Varying Input Delay
This page describes a method for constructing a full-state-feedback controller for a continuous-time system with a time-varying input delay. In particular, a condition is provided to obtain a bound on the 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  -gain of closed-loop system under time-varying delay through feasibility of an LMI. The system under consideration pertains a single discrete delay in the actuator input, with the extent of the delay at any time bounded by some known value. Moreover, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one, although results may also be attained if no bound is known. Solving the LMI for a particular value of the bound, while minimizing a scalar variable, an upper limit on the 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  -gain of the system can be shown for any time-delay satisfying this bound.


== The System ==
The system under consideration is one of the form:

  
    
      
        
          
            
              
                
                  
                    
                      x
                      ˙
                    
                  
                
                (
                t
                )
              
              
                
                =
                A
                x
                (
                t
                )
                +
                
                  B
                  
                    2
                  
                
                u
                (
                t
                −
                τ
                (
                t
                )
                )
                +
                
                  B
                  
                    1
                  
                
                w
                (
                t
                )
              
              
                t
              
              
                
                ≥
                
                  t
                  
                    0
                  
                
                ,
              
              
                0
              
              
                
                ≤
                τ
                (
                t
                )
                ≤
                h
                ,
              
              
                
                  
                    
                      τ
                      ˙
                    
                  
                
                (
                t
                )
              
              
                
                ≤
                d
                <
                1
              
            
            
              
                z
                (
                t
                )
              
              
                
                =
                
                  C
                  
                    1
                  
                
                x
                (
                t
                )
                +
                
                  D
                  
                    12
                  
                
                u
                (
                t
                −
                τ
                (
                t
                )
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{2}u(t-\tau (t))+B_{1}w(t)&t&\geq t_{0},&0&\leq \tau (t)\leq h,&{\dot {\tau }}(t)&\leq d<1\\z(t)&=C_{1}x(t)+D_{12}u(t-\tau (t))\end{aligned}}}
  In this description, 
  
    
      
        A
      
    
    {\displaystyle A}
   and 
  
    
      
        
          A
          
            1
          
        
      
    
    {\displaystyle A_{1}}
   are constant matrices in  
  
    
      
        
          
            R
          
          
            n
            ×
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n\times n}}
  . In addition, 
  
    
      
        
          B
          
            1
          
        
      
    
    {\displaystyle B_{1}}
   is a constant matrix in 
  
    
      
        
          
            R
          
          
            n
            ×
            
              n
              
                w
              
            
          
        
      
    
    {\displaystyle \mathbb {R} ^{n\times n_{w}}}
  , and 
  
    
      
        
          B
          
            2
          
        
      
    
    {\displaystyle B_{2}}
   is a constant matrix in 
  
    
      
        
          
            R
          
          
            n
            ×
            
              n
              
                u
              
            
          
        
      
    
    {\displaystyle \mathbb {R} ^{n\times n_{u}}}
  , where 
  
    
      
        
          n
          
            w
          
        
        ,
        
          n
          
            u
          
        
        ∈
        
          N
        
      
    
    {\displaystyle n_{w},n_{u}\in \mathbb {N} }
   denote the number of exogenous and actuator inputs respectively. Finally, 
  
    
      
        
          C
          
            1
          
        
      
    
    {\displaystyle C_{1}}
   and 
  
    
      
        
          D
          
            12
          
        
      
    
    {\displaystyle D_{12}}
   are constant matrices in 
  
    
      
        
          
            R
          
          
            
              n
              
                z
              
            
            ×
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n_{z}\times n}}
   and 
  
    
      
        
          
            R
          
          
            
              n
              
                z
              
            
            ×
            
              n
              
                u
              
            
          
        
      
    
    {\displaystyle \mathbb {R} ^{n_{z}\times n_{u}}}
   respectively, where 
  
    
      
        
          n
          
            z
          
        
        ∈
        
          N
        
      
    
    {\displaystyle n_{z}\in \mathbb {N} }
   denotes the number of regulated outputs. The variable 
  
    
      
        τ
        (
        t
        )
      
    
    {\displaystyle \tau (t)}
   denotes a delay in the actuator input at time 
  
    
      
        t
        ≥
        
          t
          
            0
          
        
      
    
    {\displaystyle t\geq t_{0}}
  , assuming a value no greater than some 
  
    
      
        h
        ∈
        
          
            R
          
          
            +
          
        
      
    
    {\displaystyle h\in \mathbb {R} _{+}}
  . Moreover, we assume that the function 
  
    
      
        τ
        (
        t
        )
      
    
    {\displaystyle \tau (t)}
   is differentiable at any time, with the derivative bounded by some value 
  
    
      
        d
        <
        1
      
    
    {\displaystyle d<1}
  , assuring the delay to be slowly-varying in time.


== The Data ==
To construct an 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  -optimal controller of the system, the following parameters must be known:

  
    
      
        
          
            
              
                A
              
              
                
                ∈
                
                  
                    R
                  
                  
                    n
                    ×
                    n
                  
                
              
            
            
              
                
                  B
                  
                    1
                  
                
              
              
                
                ∈
                
                  
                    R
                  
                  
                    n
                    ×
                    
                      n
                      
                        w
                      
                    
                  
                
              
            
            
              
                
                  B
                  
                    2
                  
                
              
              
                
                ∈
                
                  
                    R
                  
                  
                    n
                    ×
                    
                      n
                      
                        u
                      
                    
                  
                
              
            
            
              
                
                  C
                  
                    1
                  
                
              
              
                
                ∈
                
                  
                    R
                  
                  
                    
                      n
                      
                        z
                      
                    
                    ×
                    n
                  
                
              
            
            
              
                
                  D
                  
                    12
                  
                
              
              
                
                ∈
                
                  
                    R
                  
                  
                    
                      n
                      
                        z
                      
                    
                    ×
                    
                      n
                      
                        u
                      
                    
                  
                
              
            
            
              
                h
              
              
                
                ∈
                
                  
                    R
                  
                  
                    +
                  
                
              
            
            
              
                d
              
              
                
                ∈
                [
                0
                ,
                1
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}A&\in \mathbb {R} ^{n\times n}\\B_{1}&\in \mathbb {R} ^{n\times n_{w}}\\B_{2}&\in \mathbb {R} ^{n\times n_{u}}\\C_{1}&\in \mathbb {R} ^{n_{z}\times n}\\D_{12}&\in \mathbb {R} ^{n_{z}\times n_{u}}\\h&\in \mathbb {R} _{+}\\d&\in [0,1)\end{aligned}}}
  
In addition to these parameters, a tuning scalar 
  
    
      
        ϵ
        >
        0
      
    
    {\displaystyle \epsilon >0}
   is also implemented in the LMI.


== The Optimization Problem ==
Based on the provided data, we can construct an 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  -optimal full-state-feedback controller of the system by testing feasibility of an LMI. In particular, we note that if the LMI presented below is feasible for some 
  
    
      
        γ
        >
        0
      
    
    {\displaystyle \gamma >0}
   and matrices 
  
    
      
        
          
            
              
                P
                ¯
              
            
          
          
            2
          
          
            −
            1
          
        
        >
        0
      
    
    {\displaystyle {\bar {P}}_{2}^{-1}>0}
   and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  , implementing the state-feedback 
  
    
      
        u
        (
        t
        )
        =
        K
        x
        (
        t
        )
      
    
    {\displaystyle u(t)=Kx(t)}
   with 
  
    
      
        K
        =
        Y
        
          
            
              
                P
                ¯
              
            
          
          
            2
          
          
            −
            1
          
        
      
    
    {\displaystyle K=Y{\bar {P}}_{2}^{-1}}
  , the 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  -gain of the closed-loop system will be less than or equal to 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  . To attain a bound that is as small as possible, we minimize the value of 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
   while solving the LMI:


== The LMI: L2-Optimal Full-State-Feedback for TDS with Slowly-Varying Input Delay ==

  
    
      
        
          
            
              
              
                
                  Solve
                
                :
              
            
            
              
              
                
                
                min
                γ
              
            
            
              
              
                
                  such that there exist
                
                :
              
            
            
              
              
                
                
                  
                    
                      P
                      ¯
                    
                  
                
                ,
                
                  
                    
                      
                        P
                        ¯
                      
                    
                  
                  
                    2
                  
                
                ,
                
                  
                    
                      R
                      ¯
                    
                  
                
                ,
                
                  
                    
                      S
                      ¯
                    
                  
                
                ,
                
                  
                    
                      
                        S
                        ¯
                      
                    
                  
                  
                    12
                  
                
                ,
                
                  
                    
                      Q
                      ¯
                    
                  
                
                ∈
                
                  
                    R
                  
                  
                    n
                    ×
                    n
                  
                
                ,
                
                Y
                ∈
                
                  
                    R
                  
                  
                    
                      n
                      
                        u
                      
                    
                    ×
                    n
                  
                
              
            
            
              
              
                
                  for which
                
                :
              
            
            
              
              
                
                
                  
                    
                      P
                      ¯
                    
                  
                
                >
                0
                ,
                
                
                  
                    
                      
                        P
                        ¯
                      
                    
                  
                  
                    2
                  
                
                >
                0
                ,
                
                
                  
                    
                      R
                      ¯
                    
                  
                
                >
                0
                ,
                
                
                  
                    
                      S
                      ¯
                    
                  
                
                >
                0
              
            
            
              
              
                
                
                  
                    [
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      
                                        
                                          Φ
                                          ¯
                                        
                                      
                                    
                                    
                                      11
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          Φ
                                          ¯
                                        
                                      
                                    
                                    
                                      12
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          S
                                          ¯
                                        
                                      
                                    
                                    
                                      12
                                    
                                  
                                
                                
                                  
                                    B
                                    
                                      2
                                    
                                  
                                  Y
                                  +
                                  
                                    
                                      
                                        R
                                        ¯
                                      
                                    
                                  
                                  −
                                  
                                    
                                      
                                        
                                          S
                                          ¯
                                        
                                      
                                    
                                    
                                      12
                                    
                                  
                                
                                
                                  
                                    B
                                    
                                      1
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          P
                                          ¯
                                        
                                      
                                    
                                    
                                      2
                                    
                                    
                                      T
                                    
                                  
                                  
                                    C
                                    
                                      1
                                    
                                    
                                      T
                                    
                                  
                                
                              
                              
                                
                                  ∗
                                
                                
                                  
                                    
                                      
                                        
                                          Φ
                                          ¯
                                        
                                      
                                    
                                    
                                      22
                                    
                                  
                                
                                
                                  0
                                
                                
                                  ϵ
                                  
                                    B
                                    
                                      2
                                    
                                  
                                  Y
                                
                                
                                  ϵ
                                  
                                    B
                                    
                                      1
                                    
                                  
                                
                                
                                  0
                                
                              
                              
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  −
                                  
                                    
                                      
                                        S
                                        ¯
                                      
                                    
                                  
                                  −
                                  
                                    
                                      
                                        R
                                        ¯
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        R
                                        ¯
                                      
                                    
                                  
                                  −
                                  
                                    
                                      
                                        
                                          S
                                          ¯
                                        
                                      
                                    
                                    
                                      12
                                    
                                    
                                      T
                                    
                                  
                                
                                
                                  0
                                
                                
                                  0
                                
                              
                              
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  −
                                  (
                                  1
                                  −
                                  d
                                  )
                                  
                                    
                                      
                                        Q
                                        ¯
                                      
                                    
                                  
                                  −
                                  2
                                  
                                    
                                      
                                        R
                                        ¯
                                      
                                    
                                  
                                  +
                                  
                                    
                                      
                                        
                                          S
                                          ¯
                                        
                                      
                                    
                                    
                                      12
                                    
                                  
                                  +
                                  
                                    
                                      
                                        
                                          S
                                          ¯
                                        
                                      
                                    
                                    
                                      12
                                    
                                    
                                      T
                                    
                                  
                                
                                
                                  0
                                
                                
                                  
                                    Y
                                    
                                      T
                                    
                                  
                                  
                                    D
                                    
                                      12
                                    
                                    
                                      T
                                    
                                  
                                
                              
                              
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  −
                                  
                                    γ
                                    
                                      2
                                    
                                  
                                  I
                                
                                
                                  0
                                
                              
                              
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  ∗
                                
                                
                                  −
                                  I
                                
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                <
                0
              
            
            
              
              
                
                  where
                
                :
              
            
            
              
              
                
                
                  Φ
                  
                    11
                  
                
                =
                A
                
                  
                    
                      
                        P
                        ¯
                      
                    
                  
                  
                    2
                  
                
                +
                
                  
                    
                      
                        P
                        ¯
                      
                    
                  
                  
                    2
                  
                  
                    T
                  
                
                
                  A
                  
                    T
                  
                
                +
                
                  
                    
                      S
                      ¯
                    
                  
                
                +
                
                  
                    
                      Q
                      ¯
                    
                  
                
                −
                
                  
                    
                      R
                      ¯
                    
                  
                
              
            
            
              
              
                
                
                  Φ
                  
                    12
                  
                
                =
                
                  
                    
                      P
                      ¯
                    
                  
                
                −
                
                  
                    
                      
                        P
                        ¯
                      
                    
                  
                  
                    2
                  
                
                +
                ϵ
                
                  
                    
                      
                        P
                        ¯
                      
                    
                  
                  
                    2
                  
                  
                    T
                  
                
                
                  A
                  
                    T
                  
                
              
            
            
              
              
                
                
                  Φ
                  
                    22
                  
                
                =
                −
                ϵ
                
                  
                    
                      
                        P
                        ¯
                      
                    
                  
                  
                    2
                  
                
                −
                ϵ
                
                  
                    
                      
                        P
                        ¯
                      
                    
                  
                  
                    2
                  
                  
                    T
                  
                
                +
                
                  h
                  
                    2
                  
                
                R
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&{\text{Solve}}:\\&\qquad \min \gamma \\&{\text{such that there exist}}:\\&\qquad {\bar {P}},{\bar {P}}_{2},{\bar {R}},{\bar {S}},{\bar {S}}_{12},{\bar {Q}}\in \mathbb {R} ^{n\times n},\quad Y\in \mathbb {R} ^{n_{u}\times n}\\&{\text{for which}}:\\&\qquad {\bar {P}}>0,\quad {\bar {P}}_{2}>0,\quad {\bar {R}}>0,\quad {\bar {S}}>0\\&\qquad {\begin{bmatrix}{\begin{array}{c c c c | c c}{\bar {\Phi }}_{11}&{\bar {\Phi }}_{12}&{\bar {S}}_{12}&B_{2}Y+{\bar {R}}-{\bar {S}}_{12}&B_{1}&{\bar {P}}_{2}^{T}C_{1}^{T}\\*&{\bar {\Phi }}_{22}&0&\epsilon B_{2}Y&\epsilon B_{1}&0\\*&*&-{\bar {S}}-{\bar {R}}&{\bar {R}}-{\bar {S}}_{12}^{T}&0&0\\*&*&*&-(1-d){\bar {Q}}-2{\bar {R}}+{\bar {S}}_{12}+{\bar {S}}_{12}^{T}&0&Y^{T}D_{12}^{T}\\\hline *&*&*&*&-\gamma ^{2}I&0\\*&*&*&*&*&-I\end{array}}\end{bmatrix}}<0\\&{\text{where}}:\\&\qquad \Phi _{11}=A{\bar {P}}_{2}+{\bar {P}}_{2}^{T}A^{T}+{\bar {S}}+{\bar {Q}}-{\bar {R}}\\&\qquad \Phi _{12}={\bar {P}}-{\bar {P}}_{2}+\epsilon {\bar {P}}_{2}^{T}A^{T}\\&\qquad \Phi _{22}=-\epsilon {\bar {P}}_{2}-\epsilon {\bar {P}}_{2}^{T}+h^{2}R\end{aligned}}}
  In this notation, the symbols 
  
    
      
        ∗
      
    
    {\displaystyle *}
   are used to indicate appropriate matrices to assure the overall matrix is symmetric.


== Conclusion: ==
If the presented LMI is feasible for some 
  
    
      
        γ
        ,
        Y
        ,
        
          
            
              
                P
                ¯
              
            
          
          
            2
          
        
        x
        (
        t
        )
      
    
    {\displaystyle \gamma ,Y,{\bar {P}}_{2}x(t)}
  , implementing the full-state-feedback controller 
  
    
      
        u
        (
        t
        )
        =
        K
        x
        (
        t
        )
        =
        Y
        
          
            
              
                P
                ¯
              
            
          
          
            2
          
          
            −
            1
          
        
      
    
    {\displaystyle u(t)=Kx(t)=Y{\bar {P}}_{2}^{-1}}
  , the closed-loop system will be asymptotically stable, and will have an 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  -gain less than 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  . That is, independent of the values of the delays 
  
    
      
        τ
        (
        t
        )
      
    
    {\displaystyle \tau (t)}
  , the system:

  
    
      
        
          
            
              
                
                  
                    
                      x
                      ˙
                    
                  
                
                (
                t
                )
              
              
                
                =
                A
                x
                (
                t
                )
                +
                
                  B
                  
                    2
                  
                
                K
                x
                (
                t
                −
                τ
                (
                t
                )
                )
                +
                
                  B
                  
                    1
                  
                
                w
                (
                t
                )
              
            
            
              
                z
                (
                t
                )
              
              
                
                =
                
                  C
                  
                    1
                  
                
                x
                (
                t
                )
                +
                
                  D
                  
                    12
                  
                
                K
                x
                (
                t
                −
                τ
                (
                t
                )
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{2}Kx(t-\tau (t))+B_{1}w(t)\\z(t)&=C_{1}x(t)+D_{12}Kx(t-\tau (t))\end{aligned}}}
  with:

  
    
      
        ‖
        z
        
          ‖
          
            
              L
              
                2
              
            
          
        
        <
        γ
        ‖
        w
        
          ‖
          
            
              L
              
                2
              
            
          
        
        
          
            
              
                K
                =
                Y
                
                  
                    
                      
                        P
                        ¯
                      
                    
                  
                  
                    2
                  
                  
                    −
                    1
                  
                
              
            
          
        
      
    
    {\displaystyle \|z\|_{L_{2}}<\gamma \|w\|_{L_{2}}{\begin{aligned}K=Y{\bar {P}}_{2}^{-1}\end{aligned}}}
  will satisfy:

  
    
      
        ‖
        z
        
          ‖
          
            
              L
              
                2
              
            
          
        
        <
        γ
        ‖
        w
        
          ‖
          
            
              L
              
                2
              
            
          
        
      
    
    {\displaystyle \|z\|_{L_{2}}<\gamma \|w\|_{L_{2}}}
  Here we note that 
  
    
      
        
          
            
              
                P
                ¯
              
            
          
          
            2
          
          
            −
            1
          
        
        x
        (
        t
        )
      
    
    {\displaystyle {\bar {P}}_{2}^{-1}x(t)}
   is guaranteed to exist as 
  
    
      
        
          P
          
            2
          
        
      
    
    {\displaystyle P_{2}}
   is positive definite, and thus nonsingular.
It should be noted that the obtained result is conservative. That is, even when minimizing the value of 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  , there is no guarantee that the bound obtained on the 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  -gain is sharp, meaning that the actual 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  -gain of the closed-loop can be (significantly) smaller than 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  .
In a scenario where no bound 
  
    
      
        d
      
    
    {\displaystyle d}
   on the change in the delay is known, or this bound is greater than one, the above LMI may still be used to construct a controller. In particular, if the presented LMI is feasible with 
  
    
      
        
          
            
              Q
              ¯
            
          
        
        =
        0
      
    
    {\displaystyle {\bar {Q}}=0}
  , the closed-loop system imposing 
  
    
      
        u
        (
        t
        )
        =
        K
        x
        (
        t
        )
        =
        Y
        
          
            
              
                P
                ¯
              
            
          
          
            2
          
          
            −
            1
          
        
      
    
    {\displaystyle u(t)=Kx(t)=Y{\bar {P}}_{2}^{-1}}
   will be internally exponentially stable with an 
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L_{2}}
  -gain less than 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
   independent of the value of 
  
    
      
        
          
            
              τ
              ˙
            
          
        
        (
        t
        )
      
    
    {\displaystyle {\dot {\tau }}(t)}
  .


== Implementation ==
An example of the implementation of this LMI in Matlab is provided on the following site:

https://github.com/djagt/LMI_Codes/blob/main/L2_OptStateFdbck_cTDS.mNote that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.


== Related LMIs ==
[1] - Bounded real lemma for continuous-time system with slowly-varying delay[2] - LMI for Hinf-optimal full-state-feedback control in a non-delayed continuous-time system[3] - LMI for Hinf-optimal output-feedback control in a non-delayed continuous-time system


== External Links ==
The presented results have been obtained from:

Fridman E. 2014. Introduction to Time-Delay Systems, Analysis and Control. Springer. ISBN: 978-3-319-09392-5.Additional information on LMI's in control theory can be obtained from the following resources:

LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.


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