LQRCentralizedLTV

Sintax

[K,P] = LQRCentralizedLTV(system,T)
[K,P] = LQRCentralizedLTV(system,T,opts)

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Description

Consider a generic LTV system of the form

\[\mathbf{x}(k+1)=\mathbf{A}(k)\mathbf{x}(k)+\mathbf{B}(k)\mathbf{u}(k)\;,\]

where $\mathbf{x}(k)\in\mathbb{R}^{n}$ is the state vector, $\mathbf{u}(k)\in \mathbb{R}^{m}$ is the input vector, and $\mathbf{A}(k)$ and $\mathbf{B}(k)$ are time-varying matrices of appropriate dimensions.

Consider a standard LQR regulator

\[\mathbf{u}(k) = -\mathbf{K}(k)\mathbf{x}(k)\:,\]

where $\mathbf{K}(k)\in\mathbb{R}^{m\times n}$ is the regulator gain. Consider a regulator cost function, over a given finite window $ \{ k,\ldots,k+T \}$, where $T\in\mathbb{N}$, as

\[J(k) := \mathbf{x}^T(k+T)\mathbf{Q}(k+T)\mathbf{x}(k+T) + \sum_{\tau = k}^{k+T-1} \left(\mathbf{x}^T(\tau)\mathbf{Q}(\tau) \mathbf{x}(\tau)+\mathbf{u}^T(\tau)\mathbf{R}(\tau) \mathbf{u}(\tau)\right) \:.\]

where $\mathbf{Q}(\tau) \succeq \mathbf{0}\in\mathbb{R}^{n\times n}$ and $\mathbf{R}(\tau) \succ \mathbf{0}\in\mathbb{R}^{m\times m}$ are the time-varying weighing matrices of the state vector and control action, respectively.

The commands

[K,P] = LQRCentralizedLTV(system,T)
[K,P] = LQRCentralizedLTV(system,T,opts)

compute the well-known sequence of centralized LQR gains that solves the optimization problem

\[\begin{aligned} & \underset{\begin{subarray}{c}\mathbf{K}(\tau)\in \mathbb{R}^{m\times n} \\\tau = k,...,k+T-1 \end{subarray}}{\text{minimize}} & & J(k)\:. \end{aligned}\]

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Input arguments

Required

• system : \((T+1)\times 4\) cell array of the time-varying dynamics matrices of the LTV system, i.e.,
  • system{i,1}: \(\mathbf{A}(k+i-1),\: i = 1,\ldots, T\)
  • system{i,2}: \(\mathbf{B}(k+i-1),\: i = 1,\ldots, T\)
  • system{i,3}: \(\mathbf{Q}(k+i-1),\: i = 1,\ldots, T+1\)
  • system{i,4}: \(\mathbf{R}(k+i-1),\: i = 1,\ldots, T\)
• T : length of the finite window

Optional

• opts: struct of optional arguments (assumes default value for each parameter which is not assigned by the user)
  • verbose: display algorithm status messages (default: opts.verbose = false)

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Output Arguments

• K: \(T\times 1\) cell array of the sequence of decentralized LQR gains \(\mathbf{K}(\tau),\: \tau = k,\ldots,k+T-1\), i.e.,
  • K{i,1}: \(\mathbf{K}(k+i-1)\)
• P: \((T+1)\times 1\) cell array of the sequence of matrices \(\mathbf{P}(\tau),\: \tau = k,\ldots,k+T\), i.e.,
  • P{i,1}: \(\mathbf{P}(k+i-1)\)

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Examples

Synthetic system

To open this example execute the following command in the MATLAB command window

open LQRCentralizedLTVTutorial

Generate synthetic time-varying system matrices $\mathbf{A}(k)$, $\mathbf{B}(k)$, $\mathbf{Q}(k)$, $\mathbf{R}(k)$

T = 50;
n = 5;
m = 3;
rng(1); % Pseudo-random seed for consistency
% Alternatively comment out rng() to generate a random system
% Do not forget to readjust the synthesys parameters of the methods
system = cell(T+1,4);
for i = 1:T+1
    if i == 1
        system{i,1} = rand(n,n)-0.5;
        system{i,2} = rand(n,m)-0.5;
    elseif i == T+1
        system{i,1} = nan;
        system{i,2} = nan;
        system{i,3} = rand(n,n)-0.5;
        system{i,3} = system{i,3}*system{i,3}';
        system{i,4} = nan;
        continue;
    else % Generate time-varying dynamics preventing erratic behaviour
        system{i,1} = system{i-1,1}+(1/4)*(rand(n,n)-0.5);
        system{i,2} = system{i-1,2}+(1/4)*(rand(n,m)-0.5);
    end
    system{i,3} = rand(n,n)-0.5;
    system{i,3} = system{i,3}*system{i,3}';
    system{i,4} = rand(m,m)-0.5;
    system{i,4} = system{i,4}*system{i,4}';
end

Synthesize regulator gain using the centralized method (with some optional parameters)

opts.verbose = true;
[K,P] = LQRCentralizedLTV(system,T,opts);

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Running centralized algorithm with T = 50.
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